ScalingNumbers-Mathspool-Homepage缩放数字数学池-主页.doc

Pupil resource sheetParallel number lines: examplesPupil resource sheetParallel number linesMR1 Y8 MRScaling Numbers a b 5 8 4 5 2 516 5 2 3 4 3 6.4 22.4x ?MR3 Y8 MRa bab71 2 24361.42.1MR3 Y8 MRProportional sets or not?Identify which sets of numbers are in proportion. Then for those complete the tables below.Example 1Example 2Example 3Example 4abababaB511.25826.257497528.25242.50.75715.75311.253.512.2572.1122740122.2520400601824545.518.7586492.7ExampleScale factorInverse scale factorExampleScale factorInverse scale factorratioratioinverse as divisorfractionxxinverse as divisorfractionxxdecimalxxdecimalxxpercentagexxpercentagexxExampleScale factorInverse scale factorExampleScale factorInverse scale factorratioratioinverse as divisorfractionxxinverse as divisorfractionxxdecimalxxdecimalxxpercentagexxpercentagexxMR3 Y8 MRExampleScale factorInverse scale factorratioinverse as divisorfractionxxdecimalxxpercentagexxMR3 Y8 MRMR4 Y8 MRRatio and Proportionab357x91510.517.5a3y910.5b5201517.5Ratio and ProportionMR4 Y8 MRFind the missing numbers. Show all your working out.Example 1Example 2abab9137213.519.538.5112028.89y1015x5616Example 3a2810864b7727.5z176Example a12t19.2144b517860Between columnsa bMR4 Y8 MR4 56.8 ?Within columnsa b4 56.8 ?Pupil resource sheetMR4 Y8 MRMultiplicative relationships: key resultsFractions, decimals and percentagesWhen using a calculator, it is easiest to deal with numbers expressed in decimal form.To convert a fraction to a decimal: a/b = a b.Example: 5/8 = 5 8 = 0.625To convert a percentage to a decimal, or vice versa, remember that percentages are hundredths:Examples: 89% = 0.89, 2.75 = 275%?Scale factorsTo scale a to b, multiply by b/a.To scale b to a, multiply by a/b.?Example:5 4 45scale factor = 0.8 scale factor = 1.255 0.8 = 4 4 1.25 = 5ProportionsTwo pairs of numbers are in proportion if the scale factors for the pairs of numbers are equal.Example: a b 5 410 835 28 b/a = 4/5 = 8/10 = 28/35 = 0.8Bigger or smaller?When solving problems involving multiplication and division it is useful to ask yourself Will the answer be smaller or will it be bigger?Starting with any positive number n: Multiplying n by a number bigger than 1 increases n.Example: 0.25 5 = 1.25 (bigger than 0.25) Multiplying n by a number less than 1 decreases n.Example: 0.25 0.2 = 0.05 (smaller than 0.25) Dividing n by a number bigger than 1 decreases n.Example: 0.25 5 = 0.05 (smaller than 0.25) Dividing n by a number smaller than 1 increases n.Example: 0.25 0.2 = 1.25 (bigger than 0.25)Resource sheet PR1Proportion or not? Data setsThese examples can be used to raise the question Are these data sets in direct proportion?A13B20526287394411721418421C34D1015781220141514251630E310F771141321351667296617G411611H3945765752542962986410100I424J14.265.328486.931.74105103211476.62522455.55255.5335734Resource sheet PR2Proportion or not? Contextualised data setsThese examples can be used to raise the question Which sets of variables are in direct proportion? by considering the sets of data in context.PlaygroupThe following table is designed to show staff at a playgroup how many adults are needed to look after different sized groups of pupils.Number of adult staffMaximum number of children28316424532Although there is a constant difference in each column, the figures are not in direct proportion. The relationship could be described by c = 8(a 1).Phone billA mobile phone bill shows the following details.Calls to other networksDuration (min:sec)Cost (pence)2:3577.57:122163:049212:55387.510:103051:4452The cost is directly proportional to the duration (30p/min) but the times may need to be converted to seconds to make the relationship clear.Belt pricesA clothing website allows customers to pay in pounds sterling () or euros ()These are the prices for four different belts6.99or1113.99or2215.99or2525or39The two prices are very nearly in direct proportion. A conversion rate of 1 = 1.56 has been used, but the euro prices have been rounded to the nearest whole number.PR22 of 3Clicko kitsClicko building kits come in five sizes. Their components are listed below.KitBase PlatesLong rodsShort rodsL-jointsH-jointsBeginner110302015Designer116483224Advanced124724836Expert2421268463Supreme26018012090The number of base plates is not in proportion to the numbers of other components. However, the others are provided in the ratio 2 : 6 : 4 : 3.PhotographsA photographic shop offers to reprint enlargements of photographs according to the following table.Size of printCost10” by 8” (25cm by 20cm)5.0014” by 10” (36cm by 25cm)9.0018” by 12” (46cm by 30cm)13.80The size data are real. Several comparisons are possible. The ratio of the height to width of the different prints is not consistent (either measured in inches or centimetres). The conversion rate from inches to cm is nearly consistent at 1” = 2.5cm. The price is directly proportional to the area of the prints given in cm2.Cooling coffeeIn a science experiment, the temperature of a cup of coffee is measured over half an hour. The results are tabulated.Elapsed time in minutesTemperature in oC09857310561542203425303028The temperature is not directly proportional to the elapsed time.PR23 of 3WireA factory sells spools of wire cable by weight. The labels show the length of wire on each spool.Weight (kg)Length (m)0.75571.51143.52665380The weight and length are in direct proportion. The size of the wire can be expressed in different units, for example 76 m/kg.Beethovens symphoniesA boxed set of Beethovens nine symphonies provides the following information.Symphony numberDuration of recording in minutes136230348431529633732824964Clearly, these figures show no arithmetic relationshipWater drumA large concrete drum holds water for cattle on an Australian farm. The farmer measures the depth of the water and uses this table to estimate its volume.Depth of waterVolume0.9 m150 gallons1.2 m200 gallons1.5 m250 gallons2.4 m400 gallonsThe depth and volume measures are in direct proportion.Resource sheet PR3Proportion or not? ProblemsThese closed questions require a decision about whether the variables are in direct proportion.1 2.5 litres of paint are sufficient to cover 80 square metres. How much paint do I need to cover 250 square metres?2 A seaside harbour has a tide marker showing the depth of water inside the harbour. At midnight the depth is 4.2 m. At 2:00 am it is 4.9 m. What will the depth be at midday?3 A garage sells diesel fuel at 73.9p per litre. How much can I buy for 20?4 Henry the Eighth had six wives. How many did Henry the Fourth have?5 My recipe for 9 scones uses 200 grams of flour. How much flour will I need for 24 scones? The nine scones need 8 minutes in a hot oven. How long will I need to cook 24?6 A gardener has a lawn which is 15 m by 12 m. She decides to feed it with fertilizer applied at 1.5 grams per square metre. How much fertilizer does she need?7 A sprinter can run 100 m in 11.2 seconds. How long will it take the sprinter to run 250m?8 A shop buys cans of soft drink in boxes of 24 for 1.99 per box. They sell the cans at 39p each. Is their total profit proportional to:(a) the number of boxes they buy;(b) the number of cans they sell;(c) the money they take?9 When Robyn was 1 year old she weighed 11 kg. When she was 2 years old she weighed 14 kg. How much did she weigh when she was 4 years old?10 A 750 g box of cornflakes costs 2.19. How much does a 1 kg box cost?Resource sheet PR4Proportion or not? SituationsThese examples can be used to raise the question Are the variables in direct proportion? by considering the situation rather than sets of data.True or false?1In the different countries of the world, the number of cars on the road is directly proportional to the population.2 The weight of flour in a sack is directly proportional to the volume of the flour.3 The monthly electricity bill is directly proportional to the amount of electricity used.4 The time an audio tape plays for is directly proportional to the length of tape.5 The temperature of a saucepan of soup is directly proportional to the time it has been on the stove.6 The cost of an article of clothing is directly proportional to how long it will last.7 The time taken to read a maths problem and the time taken to solve it are in direct proportion.8 The cost of a train journey is directly proportional to the distance traveled.When could we reasonably assume the following to be true and when false?9 The time taken to drive a journey is directly proportional to the distance covered.10 The amount of money a waitress earns is directly proportional to the number of hours she works.11 The cost of a phone call is proportional to the length of the call.12 The amount of wallpaper I have to buy is directly proportional to the area of the walls I want to cover.13 The time taken to read a book is directly proportional to the number of pages in the book.L5 PaintQuestion BankYou can make different colours of paint by mixing red, blue and yellow in different proportions.For example, you can make green by mixing 1 part blue to 1 part yellow.(a)To make purple, you mix 3 parts red to 7 parts blue.How much of each colour do you need to make 20 litres of purple paint?Give your answer in litres. litres of red and litres of blue2 marks(b)To make orange, you mix 13 parts yellow to 7 parts red.How much of each colour do you need to make 10 litres of orangepaint?Give your answer in litres. litres of yellow and litres of red2 marksQuestion BankL5 Ratios(a)Nigel pours 1 carton of apple juice and 3 cartons of orange juice into a big jug. What is the ratio of apple juice to orange juice in Nigels jug?apple juice : orange juice = . : .1 mark(b)Lesley pours 1 carton of apple juice and 1 cartons of orange juice into another bigjug.What is the ratio of apple juice to orange juice in Lesleys jug?apple juice: orange juice = . : .1 mark(c)Tandi pours 1 carton of apple juice and 1 carton of orange juice into another big jug.She wants only half as much apple juice as orange juice in her jug.What should Tandi pour into her jug now?1 markTotal 3 marksQuestion BankL5 AgesThese pie charts show some information about the ages of people in Greece and in Ireland.There are about 10 million people in Greece, and there are about 3.5 million people inIreland.(a)Roughly what percentage of people in Greece are aged 40 - 59? %1 mark(b)There are about 10 million people in Greece.Use your percentage from part (a) to work out roughly how many people in Greece are aged 40 - 59. million people1 markQuestion Bank(c)Dewi says:Dewi is wrong. Explain why the charts do not show this.1 mark(d)There are about 60 million people in the UK.The table shows roughly what percentage of people in the UK are of different ages.Draw a pie chart below to show the information in the table. Label each section of your pie chart clearly with the ages.3 marksQuestion BankL6 Bids(a)One morning last summer Ravi carried out a survey of the birds in the school garden.He saw 5 pigeons, 20 crows, 25 seagulls and 45 sparrows.Complete the line below to show the ratios.1: . : . : .2 marks(b)What percentage of all the birds Ravi saw were sparrows?. %1 mark(c)One morning this spring Ravi carried out a second survey.This time he saw:the same number of pigeons25% fewer crows60% more seagullstwo thirds of the number of sparrows.Pigeons:Crows:Seagulls:Sparrows1:.:.:.3 marksL6 SaltQuestion Bank(a)What is the volume of this standard size box of salt?. cm1 mark(b)What is the volume of this special offer box of salt, which is 20% bigger?. cm2 marksThe standard size box contains enough salt to fill up 10 salt pots(c)How many salt pots may be filled up from the special offer box of salt?. salt pots1 markL6 PopulationQuestion BankEmlyn is doing a project on world population.He has found some data about the population of the regions of the world in 1950 and 1990.(a)In 1950, what percentage of the worlds population lived in Asia?Show each step in your working. %2 marks(b)In 1990, for every person who lived in North America how many people lived in Asia?Show your working. people1 mark(c)For every person who lived in Africa in 1950 how many people lived in Africa in 1990?Show your working. people1 markQuestion Bank(d)Emlyn thinks that from 1950 to 1990 the population of Oceania went up by 100%.Is Emlyn right?Tick the correct boxExplain your answer.1 markL6 ContinentsQuestion BankThe table shows the land area of each of the Worlds continents.(a)Which continent is approximately 12% of the Worlds land area?1 mark(b)What percentage of the Worlds land area is Antarctica? %2 marksL6 CurrencyQuestion Bank(a)Use 1 = 9.60 francs to work out how much 45p is in francs.Show your working.45p = . francs2 marks(b)Use 240 pesetas = 1 to work out how much 408 pesetas is in pounds.Show your working.408 pesetas = .2 marks(c)Use 1 = 9.60 francs and 1 = 240 pesetas to work out how much 1 franc is in pesetas.Show your working.1 franc = . pesetas2 marksTotal 6 marksL7 Roof framesQuestion BankTimpkins Builders make wooden frames for roofs on new houses. In the diagram of the wooden frame shown below, PQ is parallel to BC.(a)Calculate length PQ using similar triangles. Show your working. cm2 marksIn the diagram of the wooden frame shown below, angle ABC = angle LMC, and angle ACB = angle KNB.(b)Calculate length LM using similar triangles. Show your working. cm2 marksL7 PupilsQuestion BankThe table shows some information about pupils in a school.There are 408 pupils in the school.(a)What percentage of the pupils are boys?Show your working. %2 marks(b)What is the ratio of left-handed pupils to right-handed pupils?Write your ratio in the form 1: .Show your working.2 marks(c)One pupil is chosen at random from the whole school.What is the probability that the pupil chosen is a girl who is right-handed? 1 markL7 BirthsQuestion BankLook at the table:(a)In England, from 1961 to 1994, the birth rate fell by 26.1%.What was the birth rate in England in 1994?Show your working.2 marks(b)In Wales, the birth rate also fell.Calculate the percentage fall from 1961 to 1994.Show your working. %2 marksMR7 Y8 MRCurrency(a)Use 1 = 9.60 francs to work out how much 45p is in francs.Show your working.45p = . francs2 marks(b)Use 240 pesetas = 1 to work out how much 408 pesetas is in pounds.Show your working.408 pesetas = .2 marks(c)Use 1 = 9.60 francs and 1 = 240 pesetas to work out how much 1 franc is in pesetas.Show your working.1 franc = . pesetas2 marksTotal 6 marksMR7 Y8 MRL6Yoghurt(a)The label on yoghurt A shows this information.How many grams of protein does 100g of yo