Numeracy for Teaching

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Derek Haylock

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  • About the Author

    Derek Haylock is Senior Lecturer in Education at the University of East Anglia Norwich, where he is Co-Director of Primary Initial Teacher Training and responsible for the Mathematics components of the primary programmes. He has worked for 30 years in teacher education, both initial and in-service, but he also has considerable practical experience of teaching in primary classrooms. His work in mathematics education has taken him to Germany Belgium, Lesotho, Kenya, Brunei and India. He is co-author (with Anne Cockburn) of Mathematics in the Lower Primary Years (Paul Chapman Publishing, 1997), co-author (with Doug McDougall) of Mathematics Every Elementary Teacher Should Know (Trifolium Books, Toronto, 1999), co-author (with Marcel D'Eon) of Helping Low Achievers Succeed at Mathematics (Trifolium Books, Toronto, 1999) and author of Teaching Mathematics to Low Attainers 8–12 (Paul Chapman Publishing, 1991). His other publications include seven books of Christian drama for young people and a Christmas musical (published by Church House/National Society), and frequent contributions to education journals.

    Copyright

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    Read This First

    The Professional Context of Teaching

    Numeracy for Teaching is a book that, in a way, I wish I did not have to write. Of course, I am in favour of the idea that teachers, in all phases and in all subject areas, should be able to make decisions and judgements based on numerical information with confidence and a reasonable level of accuracy. Indeed, for any group of professionals, this would be a laudable aim, the achievement of which would contribute to all of us having a greater sense of general security as we go about our daily business. I am always pleased to take any opportunity to contribute to raising confidence and reducing the widespread anxiety about working with mathematical ideas that all of us who work in teacher training continue to meet in every new group of trainees. I have worked in mathematics education long enough to understand where these anxieties come from and to realise that they need a serious and sympathetic response.

    My misgivings arise from the current professional context of teaching. We now work in a field where those in control of educational policy have an obsession with reducing everything in education to numbers, tables, charts and graphs. There are people out there who actually believe that by improving the statistics you necessarily improve the quality of education. The educational ‘newspeak’ of today is littered with the language of standards, baseline assessment, target-setting, action plans, performance indicators, value-added data, levels of achievement, average points-scores, thresholds, quartiles and percentiles, Ofsted grades, audits, league tables, and so on. This, we are led to believe, is modernising the profession. Within the Department for Education and Skills (DfES, formerly the Department for Education and Employment) there is a powerful group called the Standards and Effectiveness Unit. Within that is the School Effectiveness Division and within that the Pupil Performance Team. I'm not making this up, honest! The main function of these groups seems to be to gather and disseminate huge quantities of statistical data about pupils and schools, mainly focused on levels of attainment in national tests and public examinations. As a consequence, educational achievement and practice are defined by sets of numbers. Each year this standards unit sends out to schools an ‘autumn package’ of statistical information. This is intended to enable schools to judge their performance against national standards and against the achievements of other similar schools. ‘Similar’ schools are determined by the proportions of pupils known to be eligible for free school meals, which is apparently a reliable predictor of the mix of social groups from which the school draws its population. Comparisons with the achievements of these similar schools are key factors in the judgements made by Ofsted inspectors about how well a school is doing. In some cases I have known headteachers to actively seek out the one or two more pupils they need to move them into a different band for free school meals, where the comparisons will be more favourable! This is the kind of daft thing that happens when too much significance is given to numerical indicators in judgements about the quality of teaching and learning.

    So, is this the brave new world of teaching? Of course, it isn't. The real world of teaching is still the delight of a teacher interacting with the hearts and minds of young people, the encouragement of seeing genuine learning taking place, and the occasional thrill when a pupil shows enthusiasm, flair and creativity. It continues to be about developing the skills you need to manage a class of 30 uncooperative individuals on a Friday afternoon in a mobile classroom and finding ways of making your material interesting and relevant to their needs and interests. These are the real joys and challenges of teaching.

    But the other stuff won't go away. So we just need to make sure that we can handle the mathematics, that we can make sense of the numbers and that we are not being hoodwinked by those who credit the numerical data with unjustified reliability or validity. This book is intended to make a small contribution in this respect. If you have taken the trouble to read my ramblings so far, then you will appreciate why the occasional touch of cynicism will emerge in the material that follows. I find it helps to keep me sane in today's educational climate – and I would recommend a small dose from time to time, particularly when the latest batch of DfES documents arrive at your school.

    The QTS Numeracy Test

    One of the consequences of all this is that the DfES has required the Teacher Training Agency (TTA) to introduce a Basic Skills Test in numeracy for all new entrants to the profession. It is no longer possible to achieve Qualified Teacher Status (QTS) without passing this test. The test focuses mainly on interpreting the kinds of statistics that occur in the autumn package, presumably as a way of raising awareness of this annual document within the profession. I suppose it must be a bit galling for the standards unit to keep churning out this stuff, knowing that in most schools it gets no further than a cupboard in the head-teacher's office. So, one of the main purposes of my writing this book is to provide teacher-trainees with help in preparation for the QTS numeracy test.

    Artificial Questions

    I have tried as far as possible to set the material of this book in the professional context of teaching. But, of course, I cannot provide the genuine, meaningful context in which you will encounter the need to use the numeracy skills covered in this book, nor the purposefulness in the questions which would help you to make more immediate sense of them. Just as you will find in the QTS numeracy test, many of my questions will be rather artificial. I apologise now for this, because there is plenty of research to indicate that people are much more successful with mathematics when it is purposeful and embedded in a meaningful context. So, although I have tried to draw as much as possible on the professional context of teaching, please realise that I have had to design my examples with the purpose of explaining and discussing a particular skill or concept, rather than to reflect the reality of professional decision-making.

    Sources of Data: A Disclaimer

    This book is not intended to be a source of reliable and accurate statistical data about education! As far as possible I have drawn on actual statistics and other data from individual schools, from the DfES, the QCA (Qualifications and Curriculum Authority), and their predecessor, SCAA (Schools Curriculum and Assessment Authority). I have made particular use of the data available on the DfES website. All the data drawn on in this way is therefore available in the public domain. But, occasionally, in order to make the data more accessible for teaching purposes, I have had to adapt it or prune it a little. In some cases, as will be obvious, the source of the data is my own imagination.

    I should also apologise to readers in Wales, Scotland and Northern Ireland. I work in England and I have therefore drawn on data related to the context with which I am personally familiar. I have made a definite decision not to include a few token examples from other parts of the United Kingdom to give the impression that the book draws on the full range of educational contexts. I am confident that the material will nevertheless be useful to such readers.

    Feedback from Trainees

    I have trialled the material in this book with my own students and have been encouraged by their responses:

    The material is very helpful indeed as preparation for aspects of teaching and for the QTS numeracy test. The appropriateness of the educational setting of the subject matter of each check-up adds to its usefulness.

    It is easy to follow and has clear and straightforward explanations. I found the summary of key points helpful as they are a quick reference to reinforce what has just been read. It is bound to be beneficial in preparation for the QTS skills test.

    The book is excellent. I think it will present students and others with a valuable resource, not only to help with the QTS test, but also for teaching. I could see myself using it on a dip-in-as-necessary basis.

    Many thanks for letting me work through your sample material. I cannot begin to tell you how much more I have learnt! This has definitely made me feel more confident about passing the QTS numeracy skills test!

    I have definitely demonstrated to myself from this material that my ability to complete calculations mentally has increased.

    I passed the QTS numeracy test last week after working through this material!

    Numeracy

    The mathematical material in this book focuses especially on weaknesses in numeracy that are often observed in adults in general, and in teacher-trainees in particular.

    Many adults rely too much on using calculators or formal written methods to do simple calculations that could be done mentally. So in this book I emphasise especially the development of confidence in using informal and mental methods of calculation. I am also aware that, when they do have to use calculators, many people use them inefficiently. So, help is provided in this respect as well.

    Many readers will have gained a mathematical qualification when they were 16 and not have done any formal mathematics for many years since. Understandably, they will have forgotten the meanings of some of the technical vocabulary of mathematics that is not used in everyday life, words such as median or denominator. So I have taken into account that they will need to brush up on mathematical terminology, as well as revisiting many of the basic processes and skills in which they feel a bit rusty.

    Adults are generally weak in handling the concepts of ratio and proportion. Again, many people tend to rely too much on unnecessarily formal procedures for handling problems in this area. So there is a lot of emphasis in this book on expressing proportions in fraction notation and in decimal notation, and on percentages. For example, you should be able to move freely between , 0.6 and 60%. This is an important facility when comparing the ratios and proportions that proliferate in the context of teaching.

    The biggest concern for those facing the numeracy test is likely to be the interpretation of government statistics, particularly presented in various forms of tables and graphs. By the time you have worked through this book, you should be able to handle with confidence such things as: means and modes, medians and quartiles, inter-quartile ranges, percentiles, box-and-whisker diagrams, pie charts, stacked-column charts, two-way tables, weighted means, cumulative frequency graphs, and even the DfES's favourite – value-added data.

    How to Use This Book

    This is not a book to read. It is a book to work through. You need pencil and paper to hand at all times, and, when suggested, a calculator.

    It consists of 62 check-ups, each focusing on one numeracy skill or concept. Start by trying the check-up question. The answers will be over the page. If you find the check-up question insultingly easy, then give your back a pat and your confidence a boost and go on to something else. Otherwise, work through the discussion and explanation that follows the answers. This is followed by some ‘see also’ suggestions. These will be other check-ups that might cover some of the prerequisite mathematics needed, or related areas, or extensions of the material being discussed. I then provide a summary of key points, for future reference and to highlight the main things to learn from the check-up. After this there will be one or two further practice questions. You will usually find it very helpful to work through these to reinforce and to assess your own understanding. Answers for these further practice questions are provided at the end of the book. It is important to look at these answers, because often I have included some substantial teaching points here.

    This is not a systematic book and the material does not have to be worked through in the order provided. You will probably find it most useful to dip into it from time to time whenever you have a spare half an hour or so. I assume that you will have done most of the mathematics here before. What you need is probably to revisit and practise skills and concepts from your past, to meet them again in the professional context of teaching, and to be provided with a little more enlightenment here and there about what is going on when you are manipulating and interpreting numerical data.

    And Finally…

    In common with my other books, this one will not bear the kite-mark of the Teacher Training Agency for England and Wales. I continue to turn down all invitations to submit material for this scheme, being of the opinion that intelligent readers are quite capable of deciding for themselves whether or not this book is worth buying, without it first having received the approval of a government agency.

    DerekHaylockUniversity of East Anglia, Norwich
  • Answers to Further Practice Questions

    • Multiplying by 4, ‘21 out of 25’ is equivalent to ‘84 out of 100’, or 84%. Multiplying by 5, ‘17 out of 20’ is equivalent to ‘85 out of 100’, or 85%. So, the second mark is the higher proportion of the total marks for the test.
    • For the first school, dividing by 6, ‘126 out of 600’ is equivalent to ‘21 per hundred’, or 21%. For the second school, ‘104 pupils out of 400’ is ‘52 pupils per 200’ or ‘26 pupils per 100’, i.e. 26%. The second school has the larger proportion of pupils with English as an additional language.
    • For spelling, .

      For mathematics, .

    • Unsatisfactory: .

      Good: .

      Satisfactory: .

    • School R, 80%; School Q, 74%; School P, 60%.
    • The sequence should be 1.7, 1.8, 1.9, 2.0 (or 2), …the pupil is incorrectly thinking that after ‘one point nine’ comes ‘one point ten’. The number 1.10 is the same as 1.1, which does not come after 1.9.
    • Tuesday and Wednesday have proportions greater than 0.08.
    • The target is less than 8%; Monday, 7.5%; Tuesday, 10%, Wednesday, 9%; Thursday, 7.9%; Friday, 0.9%.
      • The variable is the percentage of secondary schools whose attendance rates fall into various intervals.
      • The label 50–89% refers to those schools whose attendance rates were in the range 50–89%. The label 90–91% refers to schools whose attendance rates are in the range 90–91%, and so on. (Presumably these attendance rates have been expressed as percentages rounded to the nearest whole number.)
      • In Year 1, 17.6% of secondary schools had attendance rates in the range 92–93%.
      • 10.2% of secondary schools achieved attendance rates in the range 98–100% in Year 2. This was a higher proportion than in Year 1, where only 9.1% of secondary schools achieved attendance rates in this range.
      • It applied to 5.2% of secondary schools in Year 1 and 6.3% in Year 2.
      • 74, the total number of pupils who were absent for 5–9 days.
      • 3 were absent for 10 or more days, 34 (18+6+6+4) for less than 10 days.
      • 20 (15+4+1) were absent for 10 or more days, 245 (102+79+51+13) for less than 10 days.
      • No.
      • True.
      • False.
      • True.
      • False.
      • False.
      • This set of data could potentially contain a large number of different values, such as 5.0, 5.1, 5.2, 5.3, and so on; so it will probably have to be grouped into intervals, such as 5.0–5.2, 5.3–5.5, 5.6–5.8, and so on.
      • This variable will probably take only whole-number values from 0 to 10, so grouping will be unnecessary.
      • Point scores may take a large number of different values. An able pupil with nearly all A* and A grades in eight subjects might score around 60 points, for example. It will be necessary to group this data into intervals.
      • About 25
      • About 60
      • About 2000 classes.
      • No. The number of classes in this range is less than 1300, that is, less than 65% of the total.
    • b) and d) are continuous variables.
    • b) The stop-watch used to time the pupils will measure their times only to the nearest something. So, even though time-taken is theoretically a continuous variable, it might be rounded, for example, to the nearest tenth of a second. The times might range from about 14.5 up to maybe 24 seconds, so the rounded times could be grouped conveniently in intervals such as 14.5–15.4, 15.5–16.4, 16.5–17.4, and so on, giving about 10 groups for display in a bar chart. This is, of course, just one possible suggestion for handling the data.

      d) A very rotund teacher might have, say, a girth of 110 cm and a height of 145 cm, giving a ratio of on a calculator. By contrast, a super-model teacher might have a girth of only 50 cm but a height of 190 cm, giving a ratio of . It would be sufficient to round these ratios to two decimal places, suggesting a range of values from about 0.26 to 0.76. One way of grouping this rounded data then would be in intervals such as 0.25–0.29, 0.30–0.34, 0.35–0.39, and so on, giving possibly about 10 groups for display on a bar chart.

    • I would reply, ‘Why don't you just count them?’ However, if you are asked this daft question in a numeracy test the correct answer is 12.
    • 32.
    • The calculator result is 113.74825, so the VAT payable is £113.75.
    • Using a calculator, (approximately), . The second of these is the larger.
    • .
    • 17.5% = ‘17.5 in 100’ = ‘35 in 200’ = = (cancelling 5). So VAT is applied at the rate of £7 in every £40.
    • 12.5% is half of 25%, i.e. half of a quarter = one eighth (). 87.5% must therefore be seven-eighths (). The sum of the two fractions must be 1. It's worth memorizing these equivalents, as well as and .
      • 48% of £75 gives the same result as 75% of £48, i.e. .
      • 35% of £60 gives the same result as 60% of £35, i.e. .
    • 895 ÷ 14.5.
      • False: ‘28 sets of zero’ gives zero, so 28 × 0 = 0.
      • True: note that 28 × 0 = 0 × 28.
      • False: division by zero is meaningless.
      • True: 0 ÷ 28 could mean ‘how many sets of 28 pupils are needed to make zero pupils in total?’ Answer: zero sets!
    • 30 − (18 − 10) = 30 − 8 = 22, but (30 − 18) − 10 = 12 − 10 = 2.

      Because these results are different, subtraction is not associative. In general, A − (B − C) is not equal to (A − B) − C (unless C = 0). The situation described corresponds to 30 − (18 − 10).

    • 160 ÷ (8 ÷ 4) = 160 ÷ 2 = 80, but (160 ÷ 8) ÷ 4 = 20 ÷ 4 = 5.

      Because these results are different, division is not associative. In general, A ÷ (B ÷ C) is not equal to (A ÷ B) ÷ C (unless C = 1 or −1).

    • 28 × 25 = (7 × 4) × 25 = 7 × (4 × 25) = 7 × 100 = 700. Cost = £700.
      • (100 + 80) × 8 = (100 × 8) + (80 × 8) = 800 + 640 = 1440.
      • (200 − 20) × 8 = (200 × 8) − (20 × 8) = 1600 − 160 = 1440.

      Cost = £1440.

      • First find the total cost of one each of the two books, £12+£4 = £16, then multiply this by the number of pupils: 25 × £16 = £400.
      • First find the total cost of textbooks (25 × £12 = £300) and the total cost of workbooks (25 × £4 = £100), then add these: £300+£100 = £400.
      • (£700 +£630) + 7 = (£700 + 7) + (£630 ÷ 7) = £100 + £90 = £190.
      • (£1400 − £70) + 7 = (£1400 + 7) − (£70 + 7) = £200 − £10 = £190.
      • Doing the operations in the order entered, the four-function calculator would give the result as 3.
      • Giving precedence to the division, the scientific calculator would give the result as 9.
    • The operations have been done in the order entered. This is using a basic four-function calculator. The result displayed is correct in this context.
    • My estimate was around £84 (£20 + £9 + £15 + £36 + £4).

      Actual cost = £85.75.

    • C looks most likely. We need less than £2 for each of less than 70 pupils, so the cost should be under £140. Actual cost is £131.92.
    • The calculator result of 181.7 has been misinterpreted as £181.07. It should be £181.70.
    • Calculator sequence: MRC, MRC, 4.15 × 5 =, M+, 2.95 × 3 =, M+, 5.45 × 3 =, M+, 3 × 12 =, M+, 0.95 × 4 =, M+, MRC.
    • 10.
    • £554 869.
    • (no calculator required)

      to one decimal place

      (no calculator required)

      to one decimal place

      to one decimal place.

    • Females: C, 19.5%; D, 18.3%, E, 14.2%; N, 8.2%.

      Males: C, 18.5%, D, 17.9%, E. 14.0%; N, 8.5%.

    • Giving these to the nearest whole percent would not discriminate sufficiently between the data. For example, both females and males would have 18% for grade D. It would also exaggerate some differences, e.g. giving one whole percent difference between females and males in the N-grade category (using 8% and 9% respectively, instead of 8.2% and 8.5%).
    • To order these populations, look first at the power of 10 and then at the digits. The order is: UK, Japan, India, China.
    • £290 000 000 or £2.9 × 108.
    • 4.4820717 × 10−3 = 0.0044820717 = approximately 0.4%.
    • One method is to round the £4.95 up to £5. Then 24 × £5 = £120. Subtract 24 × 5p = £1.20. Answer £118.80.
    • One method is to use factors, writing 48 as 4 × 12. Then 125 × 4 = 500 and 500 × 12 = 6000. Answer £6000.
    • The 97 is close to 100, so think of it as 100 − 2 − 1. Then we need (240 × 100) − (240 × 2) − (240 × 1) = 24 000 − 480 − 240 = 23 520 − 240 = 23 280.
    • 45 × 74 = 45 × 2 × 37 = 90 × 37 = 100 × 37 − 10 × 37 = 3700 − 370 = 3330

      Area = 3330 square metres.

      44×75 = 11 × 2×2×75 = 11×2×150 = 11× 300 = 3300

      Area = 3300 square metres, which is smaller.

    • You could start by dividing both numbers by 3 to give 48 ÷ 3 = 16. Or you could think of the 144 as 180 − 36, which when divided by 9 gives 20 − 4 = 16. Or you could break the 144 up into 90 + 54, giving 10 + 6 = 16.
    • 6035 ÷ 85 = 12070 ÷ 170 (doubling both numbers) = 1207 ÷ 17 (dividing by 10). To divide 1207 by 17, I would start with 1700 (100 × 17), which is 493 too much. The 493 can be split up into 340 + 153 = 340 + 170 −17, each bit of which can be divided easily by 17. Answer: 100 − (20 + 10 − 1) = 71, i.e. £71 per pupil.
    • For 893 ÷ 24, I would start with 30 × 24 = 720. So I need another 173. Next, 5 × 24 = 120, so I need another 53. Then, 2 × 24 = 48, which leaves me just 5 short. The result is 30 + 5 + 2 = 37, with 5 remainder. This 5 is less than half a mark per pupil, so to the nearest whole number the average mark is 37.
      • 12.5% of 160 = of 160 = 20;
      • 30% of 220 = of 220 = 66.
    • 50% = 120; 25% = 60; 1% = 2.4; 2% = 4.8. Adding these, 78% = 187.2. So about 187 pupils, or 188 to pass the target.
    • The percentage not reaching level 4 is 100% − 40% − 36% = 24%. So we need 24% of 125. which is 25; which is 5. Total = 30.
    • Key sequence for first method: 279 × 47.8 ÷ 100 =

      Key sequence for second method: 279 × 47.8 % (procedure not universal) Key sequence for third method: 0.478 × 279 =

      The result is 133.362, so to surpass this percentage 134 pupils must achieve the required grades.

    • Stepping from 14.87 to 14.9 to 15 to 20 and then to 100, the difference between 14.87% and 100% is 0.03 + 0.1 + 5 + 80 = 85.13%.
    • 2.970 + 34.000 + 1.085 = 38.055 m.
    • 3.620 − 2.085 = 1.535 m.
    • 132.
    • 330.
    • The areas method gives four multiplications: 20 × 30, 20 × 9, 8 × 30, 8 × 9, giving 600 + 180 + 240 + 72 = 1092 hours.
    • The areas method gives six multiplications: 100 × 70, 100 × 2, 40 × 70, 40 × 2, 2 × 70, 2×2, giving 7000 + 200 + 2800 + 80 + 140 + 4 = 10 224 square metres, just larger than a hectare.
    • I subtracted first 10 classes of 28 (280), leaving 364, then another 10, leaving 84, then 2 classes of 28 (56), leaving 28, which was 1 more class. In total this gave 10 + 10 + 2 + 1 = 23 classes.
    • I subtracted first 20 coach-loads of 42 (840), leaving 710, then another 10 (420), leaving 290, then 5 coach-loads of 42 (210), leaving 80, which was 1 more coach-load with 38 remainder. In total this gave 20 + 10 + 5 + 1 = 36 coach-loads, with 38 remainder. So, 37 coaches are needed.
    • 1 × 2 × 3 × 4 × 5 = 120 → 0.1 × 0.2 × 0.3 × 0.4 × 0.5 = 0.00120 (with five figures after the point). This can be written as 0.0012, but don't drop the final zero until after you have decided where to put the point.
    • 0.6 × 0.12 → 6 × 12 = 72 → 0.072.
    • 12% on FSM, 60% of these, no adult in employment.

      60% of 12% = 7.2% (50% of 12 is 6, 10% of 12 is 1.2, add these to get 60% of 12).

    • 0.71 × 0.71 = 0.5041, so the area of the paper is reduced by a factor of about 0.5. This is consistent with the fact that a sheet of A5 paper is half of a sheet of A4.
    • 0.126 ÷ 0.09 = 126 ÷ 90 = 14 ÷ 10 = 1.4 or 140%.
    • a) is the same (both numbers doubled), c) is the same (both multiplied by 100), and d) is the same (both divided by 10).
      • 2
      • 0.2
      • 200
      • 0.002
    • B: the actual cost is £315.21.

      Calculators will cost about £120, transparencies about £24, envelopes about £8, which is £152 in total. VAT is less than a fifth of this, say, a bit less than £30, making about £180 in total. Add about £135 for the dictionaries (I used 15 × £9), giving about £315 in total.

    • To the nearest percent: 93%, 57%, 33%, 34%, 12%. Estimates within 2% either side of these are very good.
    • One possibility is 08.55–09.05 registration, 09.05–09.25 mental arithmetic, 09.30–10.20 mathematics, 10.45–11.00 free time, 11.00–11.55 English.
    • Taught-week = 22.5 hours, 1 hour short of the recommended 23.5 hours.
    • 1189 mm = 118.9 cm = 1.189 m.
      • about 17.5 cm,
      • about 20 m,
      • about 20 mm.
    • 70 mph is just over 110 kilometres per hour, so a journey of about 10 000 km will take around 90 hours. Since you also travel through 90° of latitude on this journey, this means that when you are travelling due south (or north) at 70 mph you are moving through about 1° of latitude per hour.
    • A5 paper is approximately 149 mm by 210 mm, or 0.149 m by 0.210 m. These dimensions give the area as approximately 0.149 × 0.21 = 0.03129 m2. The fraction as a decimal is 0.03125.
    • The volume is 0.175 × 0.095 × 0.065 = 0.0010806 m3, which is just over 0.001 m3, or one thousandth of a cubic metre, or 1000 cm3. (NB: 1 litre = 1000 cm3.)
    • 1 ha = about acres. 0.4 ha = 4000 m2, which could be 40 m × 100 m, or 80 m × 50 m, 160 m × 25 m, and so on.
    • £6.50 for half a litre is £13 per litre. £5 for 400 ml is £1.25 for 100 ml, so £12.50 for 1 litre. On this basis, the second is the better buy.
    • 1.25 dl = 0.125 litres = 125 ml = 25 medicine-spoonfuls.
      • a quarter of a pound,
      • 330 ml,
      • 40 litres,
      • 70 kg (which is about 11 stone).
    • An A4 sheet is of a square metre in area (half of A3, which is half of A2, which is half of A1, which is half of A0, which is 1m2 in area). So its weight is of 80 g = 5g. A ream is 500 sheets, so weighs 2500 g = 2.5 kg. You can safely put 8 sheets of standard A4 paper (40 g) in an envelope (less than 20 g) and stay within the 60-g limit. This is a useful piece of knowledge!
    • 50 Swiss francs per pupil is 2100 Swiss francs in total.
    • The teacher can buy 14 marker pens, costing £17.50 in total. My mental starting point was 4 for £5.
    • The total cost is £556.80, which is £560 to the nearest £10. I assumed that I would not be far out if I let the lapel badge cost 25p. That's £1.75 per pupil, £3.50 for 2, £7 for 4. Multiply this by 80, to get £560 for 320 pupils. This is obviously £3.20 over the actual cost.
    • The teacher has enough in the budget to buy 31 textbooks. I started this mental calculation by thinking that 30 at £13 would be £390. So, 30 at £12.90 would be £3 less than this, i.e. £387. That leaves £13, enough to buy one more textbook.
    • The difference is £24,500. The ratio is 42 000:17 500, which could be simplified in various ways, such as 420:175 = 84:35 = 12:5 = 24:10. This is 2.40:1, or £2.40 for every £1.
    • The difference is now £25,382. But the ratio is still 2.40:1. Notice that applying a percentage-increase increases the difference, but leaves the ratio the same. Those on higher salaries would prefer this kind of pay rise.
    • The difference is still £25,382. But the ratio is now 2.36:1. Notice that applying a flat-rate increase of this kind leaves the difference the same, but reduces the ratio. Those on lower salaries would prefer this kind of pay rise.
    • The £4800 must be divided by 8 (1 + 2 + 5), giving £600. The allocations are £600, £1200 and £3000 for nursery, infant and junior respectively.
    • Girls: approximately. Boys: approximately. Number of girls = 41 000 ÷ 7 × 4 = about 23 400. Number of boys = 41 000 ÷ 7 × 3 = about 17 600.
    • 61%, 58.8%.
    • 6543 × 1.21 and 6543 × 0.79.
    • It makes no difference! The price in both cases is £815.83.

      VAT first is: 789 × 1.175 × 0.88. Reduction first is: 789 × 0.88 × 1.175.

    • The increase is £150, which is 12% of £1250.

      After decreasing £1400 by 12% we should expect the answer to be less than £1250, because the reduction is 12% of £1400, whereas the previous increase was 12% of only £1250.

      Calculator result: £1400 × 0.88 = £1232, which, as predicted, is less than £1250.

    • Science increases by 0.2; 0.2 ÷ 31.6 = 0.006 (approximately) = 0.6%.

      Mathematics decreases by 1.8; 1.8 ÷ 32.9 = 0.055 (approximately) = 5.5%.

    • The proportion increases by 5 percentage points (54.4 − 49.4). Expressed as a percentage of the starting value, 5 ÷ 49.4 = 0.101 (approximately) = a 10.1% increase over the five years.

      The biggest annual percentage increase was 3.7% from 1998 to 1999.

    • Note that the increase is not 26 percentage points, but 26% of the previous proportion. So 126% (of the previous proportion) = 63 (i.e. 63% of the pupils), which gives 2% = 1, so 100% = 50. So the previous proportion was 50%.
    • 85.8% = 37 900, so 1% = 37 900 ÷ 85.8, and 100% = 37 900 ÷ 85.8 × 100. This gives 44 172.494, which is 44 200 to the nearest hundred.
    • The mean class-size for School X is 28.6; the mean class-size for School Y is 27.1. These might be calculated to compare the schools with some national data about mean class-sizes. Or, to see whether, on the whole, the schools are achieving some target for reduction of class-size; but the mean of 28.6 for School X will be little consolation for the teacher with a class of 35! All other things being equal (which is highly unlikely), a comparison might be made between the mean class-sizes of the schools as part of an evaluation of the impact of class-size on pupil achievement.
    • The total number of pupils is 129. To find the total number of points, note that there are sixteen 21s, thirty-two 27s, and so on. So, the total is found by calculating (16×21) + (32 × 27) + (40 × 33) + (25 × 39) + (8 × 45) + (6 × 51) + (2 × 57) = 4275. So the mean is 4275 ÷ 129 = 33.1 to one decimal place. Given all the other variables involved, this is a fairly meaningless statistic. It assumes, for example, that a school should get the same credit for two pupils gaining levels 4 and 8 respectively as for two pupils both gaining level 6. There is no valid basis for such an assumption.
    • The mode is level 4.
    • The mode was level 5.
    • The modal interval is £4.00-£5.99.
    • This school is in the ‘FSM more than 50%’ group. Compared to schools in this group, the results in terms of A*–C grades for GCSE mathematics are ‘better than average’, because their 28% is higher than the median of 18%; i.e. this school did better than at least half of the schools in this group.
    • St Anne's devotes 22.1% of the Y3 teaching week to English. This is less than the median percentage for all primary schools. More than half of all schools devote a larger percentage of the Y3 teaching week to English than does St Anne's.
    • Compared to schools in this FSM group, with 28% achieving grade C or above in mathematics, the first school has a proportion higher than the UQ (25%). With 15% achieving this level, the second school has a proportion lying between the LQ (12%) and the median (18%). Loosely speaking, the first school has done well compared to similar schools, being in the top quarter based on the proportions of pupils gaining grade C or above in mathematics. The second school is only ‘fairly average’, below the median but not in the bottom quarter of these schools.
    • The percentage of the Y3 teaching week for English at St Anne's (22.1%) is just less than the LQ (22.2%) for all primary schools. This puts them in the bottom quarter of schools in terms of the proportion of the Y3 teaching week devoted to English. The proportion for St Michael's (30%) exceeds the UQ (28.1%) for all primary schools, putting them in the top quarter for this variable. Loosely speaking, St Anne's has a low proportion of the Y3 week devoted to English, whereas St Michael's has a high proportion.
    • The ‘average’ numbers of hours for RE and PE given by the median values are very similar. However, there is much more variation between schools in the time given to PE than there is in the time given to RE. The range for PE (1.7 hours) is greater than that for RE (1.3 hours). Also, the IQR for PE (0.7 hours) is greater than that for RE (0.4 hours), suggesting that the greater variability is not just due to a few schools giving an exceptionally high number of hours to PE.
    • A is valid. The maximum marks achieved were 91 and 100 for literacy and numeracy respectively.

      B is invalid. The median marks were about 40 and 60 for literacy and numeracy respectively.

      C is invalid. The diagram does not tell us anything about how individual pupils did in the tests.

      D is invalid. A mark of 40 in literacy was bettered by about 75% of the pupils, whereas 40 in numeracy was the median mark.

      E is valid. A mark of 70 in literacy is in the ‘fairly average’ box, whereas a mark of 70 in numeracy is in the ‘high-scoring’ whisker.

      F is probably valid. The numeracy box (the middle 50% of pupils) and the bottom whisker (the bottom 25%) are substantially lower than the literacy box and lower whisker.

      G is invalid. It should be the lowest 50 scores, i.e. the bottom 25%.

      H is valid. The top whisker for numeracy is much longer than that for literacy.

      • The bottom plot refers to ‘all schools in England with Y6 pupils with 8% or less pupils eligible for FSM’ and the top plot refers to ‘all schools in England with Y6 pupils with more than 50% of pupils eligible for FSM’. The variable in both cases is the percentage of pupils achieving level 4 or above in the Key Stage 2 English national assessment that year.
      • Based on the percentages of pupils achieving level 4 or above in Key Stage 2 English, the top 25% of schools in the group with 8% or less FSM had proportions of pupils achieving this level ranging from about 93% to 100%. Based on the percentages of pupils achieving level 4 or above in Key Stage 2 English, the middle 50% of schools in the group with more than 50% FSM had proportions of pupils achieving this level ranging from about 46% to 67%.
      • There is a very striking contrast between the performances of the two groups. The boxes (the middle 50%) do not even overlap. All the schools in the first group scored higher in terms of the variable than the median for the second group. To put this another way, more than half the schools in the second group had a lower proportion of pupils achieving this level for English than the lowest proportion achieved by any school in the first group (60%). And the highest proportion achieved by any school in the second group (87%) was equal to the median for the first group and was therefore equalled or surpassed by at least half of the schools in that group.
      • Hay is in the bottom 25% of schools in their FSM group and has therefore done poorly in English compared with similar schools. Lock is in the top 25% of schools in their FSM group and has therefore done well in English compared with similar schools.
      • The top 40% of schools, based on their Key Stage 1 results for reading, had average points scores of 16.2 or more; the bottom 60% had average points scores of 16.2 or less.
      • Clarendon Primary was in the top 40% based on average points scores for Key Stage 1 reading, but not in the top 25%. But for writing they were in the top 5%.
      • At least 5% of the schools, which is about 800 schools or more.
    • If all the non-selective secondary schools in this FSM group were ranked according to the proportions of pupils gaining grade C or above in GCSE science, then a school coming nine-tenths of the way along the line would have about 69% of their pupils gaining these grades in science. In other words, the top tenth of schools would have 69% or more of their pupils gaining these grades in science.
      • 5
      • 10
      • 2
      • 28.
      • 52% for boys and 72% for girls (this was actually for Art and Design) – a difference of 20 percentage points.
    • The points are generally clustered around a rising diagonal line, indicating a positive correlation. In regions C, E and G there are 4, 6 and 2 points, giving 12 out of 20 points in total; there are no points in regions A and J. This confirms a positive correlation.
      • A positive correlation might be expected. Older pupils would tend to be taller.
      • No correlation would be expected. I know of no reason why the size of the head should be related to the score in a numeracy test.
      • A negative correlation might be expected. The pupils living nearest would tend to leave home later; those living further away would tend to leave home earlier.
    • £87.50.
      • This is not direct proportion: a letter of 40 g will not cost twice as much as one of 20g, for example.
      • This is not direct proportion. A square of side 5 m has an area of 25 m2; a square of side 10 m has an area of 100 m2; the ratios 5:25 and 10:100 are not equal.
      • This is direct proportion. Weights in pounds to weights in kg are always in the same ratio, about 1:0.454.
      • This is not direct proportion, although it is sometimes difficult making parents understand this. A pupil at level 2 in Year 2 is unlikely to be at level 4 in Year 4, level 6 in Year 6 and level 8 in Year 8!
    • (0,0), (254,100), (127,50)…

      The ratio is about 1:0.39 (using a calculator), so the gradient will be about 0.39. This means that 1 cm is about 0.39 of an inch.

    • Only (b) might be sensible, but this is assuming that each pupil can be located in one and only one ethnic origin group and that there are not too many different groups.

      A pie chart could not be used for (a) because there is not a single population to be represented by the pie. A pie chart would be hopeless for (c) because there are 52 subsets!

      • third
      • half
      • walking.
      • bus (because 48 pupils = 20%).
      • About 61.
      • About 28.

      The transposition of the digits here make these two results easy to remember as reference points for converting temperatures.

      In (a) the 9C/5 must be calculated before adding on 32.

      In (b) the (F − 32) bracket must be done first, before multiplying by 5 and dividing by 9.

      • L = (2A + 5B + 2C + D)/10.
      • (2 × 3 + 5 × 4 + 2 × 3 + 3)/10 = 35 ÷ 10 = 3.5, which rounds up to level 4.
      • (2 × 5 + 5 × 4 + 2 × 5 + 4)/10 = 44 ÷ 10 = 4.4, which also rounds to level 4.

      Try explaining that to the parents! This is the kind of daft thing that happens when you reduce educational performance to a set of numbers.

    • (5.6 × 19 + 7.8 × 20 + 8.1 × 22 + 4.4 × 5)/(19 + 20 + 22 + 5) = 7.01.
    • You need to know how many pupils are in each of the three stages. Then find the weighted mean of the three percentages, using these numbers as weightings.
      • ‘60% of schools with 8% or less of their pupils eligible for free school meals had up to and including 40% of their pupils achieving level 3+ in the Key Stage 1 reading test.’
      • About 21% (100% − 79%).
      • The Ford School's percentage of pupils gaining level 3+ for reading equalled or bettered that of about 70% of schools in this FSM group.

      This example of a cumulative frequency graph (not untypical of government statistics) is difficult to read, because it contains percentages used in three different ways: the percentage of pupils eligible for free schools meals (FSM), the percentage of pupils gaining level 3+ in reading, and the percentages of schools with various proportions of pupils achieving this level.

    • The numbers below are approximate:

      A score of 82% is above the 95th percentile, i.e. in the top 5% for the year group.

      • Teacher assessments have been fairly consistent, while test results have fluctuated markedly by comparison.
      • While the proportion achieving level 5+ by TA did not change much, there was a dramatic rise in the proportions reaching this level in the test. Explanations: teachers got better at preparing pupils for the test? The test got easier?
      • The proportions assessed at level 5+ by TA and by the test coincided.
      • By not starting at zero on the vertical axis a misleading impression can be given of the rate of change from one year to the next.
    • Only (c) is a variable changing with the passing of time, which would appropriately be represented in a line graph.
    • The first purpose of the graph is to compare the total numbers of pupils per A-level subject, shown by the heights of the columns. The second purpose is to show the contributions of boys and girls to these totals. For example, music is clearly the lowest frequency of these five subjects, but within this the contribution of the girls can be seen to be far greater than that of the boys.
      • ‘14 is less than or equal to the Key Stage 1 average points score, which is less than 16’: this means an APS from 14 up to but not including 16. Most pupils in this group (66%) achieved level 4 in Key Stage 2 maths.

        ‘Key Stage 1 average points score greater than or equal to 18’. Most pupils in this group (65%) achieved level 5 in Key Stage 2 maths.

      • Key Stage 1 APS = 13. Only 6% of pupils in this group achieved level 5 or above in Key Stage 2 maths.
      • 64
      • Any score from 52 to 62; 57 is bang in the middle.
      • 50
    • C, of course!

    Sources of Data

    Below is a list of the published sources of the data used in the check-ups and further practice questions in this book.

    • The Department for Education and Employment Autumn Packages for Schools, 1998, 1999, 2000, available at the DfES website:

      http://www.standards.gov.uk/performance

      Check-ups 04, 11, 24b, 25a, 31b, 37c, 38c, 40c, 44, 45, 47, 48, 49, 54, 57, 58, 59, 61, 62

      Further Practice 11.1, 31.2, 38.2, 40.3, 42.2, 44.1, 44.2, 45.1, 45.2, 48.1, 49.1, 49.1, 50.1, 57.1, 59.1, 61.1, 62.1

    • Other sections of the DfES website:
    • Qualifications and Curriculum Authority (1999) Standards at Key Stage 2, English, Mathematics and Science, report on the 1998 National Curriculumn assessments for 11-year-olds. Sudbury: QCA Publications

      Further Practice 20.3

    • Qualifications and Curriculum Authority (1999) Standards at Key Stage 3, Mathematics, report on the 1998 National Curriculum assessments for 14-year-olds. Sudbury: QCA Publications

      Check-up 21

      Further Practice 22.1

    • Qualifications and Curriculum Authority (2000) Standards at Key Stage 2, English, Mathematics and Science, report on the 1999 National Curriculum assessments for 11-year-olds. Sudbury: QCA Publications

      Check-ups 20b, 24a

      Further Practice 25.1

    • Qualifications and Curriculum Authority (2001) Assessment and Reporting Arrangements Key Stage 2. Sudbury: QCA Publications

      Further Practice 55.1

    • School Curriculum and Assessment Authority (1996) GCSE Results Analysis. London: SCAA Publications
    • School Curriculum and Assessment Authority (1996) GCSE Results Analysis. London: SCAA Publications

      Check-up 30a

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