Progress in Understanding Mathematics Assessment (PUMA)

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1 Progress in Understanding Mathematics Assessment (PUMA) Interim Manual for Autumn tests Years 3 to 6 Colin McCarty & Caroline Cooke

2 Copyright 2014 Hodder and Stoughton Ltd. Photocopying is prohibited. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, without permission in writing from the publisher. This publication is excluded from the reprographic licensing scheme administered by the Copyright Licensing Agency Ltd and may not be photocopied. Printed in England for Hodder Education, part of Hachette UK, 338 Euston Road, London NW1 3BH. 2

3 Contents 1 Introduction 04 This interim manual 04 What is PUMA? 04 PUMA Curriculum Maps 04 Why use PUMA? 05 What does PUMA provide? 05 2 Administering the PUMA Tests 06 When to test 06 How to test 06 Group size 06 Timings 06 Preparation 06 Test conditions 06 Administration 06 3 Answers and mark schemes 07 Marking and recording results 07 PUMA 3 Autumn 08 PUMA 4 Autumn 10 PUMA 5 Autumn 12 PUMA 6 Autumn 15 4 Obtaining and interpreting test s 18 Summative measures 18 Reporting progress the PUMA Scale 20 Predicting future performance 22 5 Standardised conversion tables 24 PUMA 3 Autumn 24 PUMA 4 Autumn 25 PUMA 5 Autumn 26 PUMA 6 Autumn 28 3

4 1 Introduction This interim manual for PUMA Autumn We have brought forward publication of the PUMA Autumn tests, so that you can start assessing Mathematics in the Autumn term, as you start using the new National Curriculum. To support you in using the PUMA Autumn tests, we have published this free teacher guidance which provides everything you need to administer and mark the Autumn tests. More extensive teacher guidance, including information relating to the PUMA Spring and Summer tests will be provided in the full PUMA Manuals Stages 1 and 2, which will be published in March 2015, together with the PUMA tests for Spring and Summer. The PUMA Manuals will also include the following information, to assist you when using PUMA across the whole school year: Diagnostic and formative information Pupil profile sheets for each term, to enable you to review patterns of strengths and weaknesses across the year. Further information about interpreting and analysing results Technical information about the standardisation Teacher scripts and mark schemes for the PUMA Spring and Summer tests. In the meantime, should you have any queries about using the PUMA Autumn tests, please What is PUMA? Progress in Understanding Mathematics Assessment (PUMA) is a suite of tests written to the new National Curriculum. PUMA is designed to be used toward the end of each term, to measure and monitor pupils progress term by term, providing reliable, predictive and diagnostic information. The autumn test is a wider span test than the more focused Spring and Summer tests, with questions relating to mathematics covered in earlier years; it should be used to baseline the children (who will have only had at best one term of teaching on this year s curriculum). PUMA is designed for whole-class use, with pupils of all abilities. The tests are easy and quick to administer each taking between minutes, depending on the year and are straightforward to mark. PUMA Curriculum Maps The PUMA tests provide thorough coverage of the new National Curriculum Programme of Study for the particular year. We have created Curriculum Maps, breaking down the Programme of Study for the year term by term. These Curriculum Maps help to define what PUMA assesses each term. The Curriculum Maps can be downloaded from Schools taking part in the standardisation of PUMA followed these Curriculum Maps to guide them in delivering the new National Curriculum, before it was statutory. This made the standardisation of the PUMA tests a valid assessment of the new National Curriculum. 4

5 Why use PUMA? PUMA provides reliable summative information. For example: PUMA uniquely provides three carefully designed tests for each year, enabling you to follow the progress of your pupils from term to term, as well as year to year throughout primary school. Marks have been calibrated onto the PUMA Scale to allow you to follow progress term by term and compare progress to national norms see page 20 for further details. It allows you to predict what they should obtain in subsequent terms and so set meaningful targets. However, if you need to establish a National Curriculum level for each pupil, PUMA tests are calibrated to indicate National Curriculum levels. Also, because it has a diagnostic capability, PUMA enables you to investigate some of the strengths and weaknesses of your pupils mathematics skills. To enable you to use the information in a diagnostic/formative way, total s can be broken down into distinct aspects of mathematics, giving a useful profile which reflects the categories of the new National Curriculum. These are: Number, place value and rounding Addition, subtraction, multiplication and division; algebra Fractions, decimals and percentages, ratio and proportion Measures Geometry: shapes position, direction, motion Statistics, data handling. What does PUMA provide? PUMA provides a standardised assessment of a pupil s mathematics attainment, plus a profile that helps you to identify pupils who may need further teaching and practice, as well as helping you to identify where they are doing well. It provides four global measures of mathematics attainment for each pupil: an age-standardised (from which a percentile can be derived) a Mathematics age a on the PUMA Scale National Curriculum sublevels and APP level. Each test also gives a points (widely used by local authorities). The PUMA test results have been statistically linked from term to term and year to year to enable you to track or predict progress through the whole primary phase. This also enables detailed comparison of individual patterns of performance against the norms and patterns for the term. Underpinning all this is the PUMA Scale: this gives the equivalent of a decimalised level which enables you to monitor small increments of progress from term to term. Although National Curriculum levels are no longer in use, these tests carry forward the standard from 2014, so that you have a measure that may be compared back to previous years, at this time of transition. The PUMA Scale acts as a common spine on which all the PUMA tests across the primary phase are plotted (Table 4.3 on page 20 draws this all together). It provides the statistical basis for predicting pupil progress and future attainment, based on the termly performance data of over 10,000 pupils nationally. 5

6 2 Administering the PUMA tests When to test The PUMA tests have been designed to assess the National Curriculum objectives presented in the PUMA Curriculum Map for that term. They should, ideally, be used just before the end of term. How to test Give each pupil a test booklet and ask them to write their names on the front cover. Before the test, tell pupils these key points: That pupils need to read the questions themselves, but weak readers may be given help with reading the question (see Administration below). There will be some sections they can do easily, particularly the earlier questions. They shouldn t worry if they find some questions difficult. They should just try their best and move on to see if they can answer some of the following questions. Ask them to write answers clearly. If they change their mind, they should cross or rub out the wrong answer and write in the new answer. Group size You can administer the tests to a whole class or large group, if you feel comfortable doing so. With weaker Year 3 children, however, it may be better to work with small groups, with the TA also delivering the test for example, five or six children of similar ability so that pauses can be taken, if required. Timings A maximum time limit of 50 minutes is advised for the Year 3 and 4 tests, and 60 minutes for the Year 5 and 6 tests. In the PUMA standardisation we found that it took well under one minute for a mark for most pupils, unless they were particularly hesitant or slow workers, where extra time may be allowed. Preparation Each pupil will need the appropriate test booklet and a pencil or pen. Test conditions For results to be reliable, it is important that the pupils work alone, without copying or discussing their answers. Administration If any pupils are uncertain about what they need to do, you may give additional explanation to help them understand the requirements of the test. Do not, however, help with the mathematical content of the question or read out any of the actual numbers that form part of the question. 6

7 3 Answers and mark schemes Marking and recording results Use the box in the right-hand margin alongside each question in the test booklets to record marks. Some questions have more than one part, or attract more than one mark, so please follow the mark scheme carefully. No half-marks should be awarded. Beneath each box there are code letters indicating the category of maths the question focuses on (using abbreviations for numeracy, operations, fractions, geometry, measures, statistics and problem solving). If you would like to profile the pupil s performance, add up the number of marks they have obtained in each coded category and record them on the front cover of the test booklet. You can record total marks for the page at the bottom of each page in the test booklets. Then add together the page s to find each pupil s total raw and record this at the bottom of the front cover. Please use your professional judgement when marking. For example, accept mirror reversals of single digits but not reversals of double-digit numbers. Equally, any clear indication of the answer is acceptable, irrespective of what was asked for (e.g. an object ticked instead of circled). When you have calculated the total raw for each pupil, refer to the conversion tables to obtain: the age-standardised Mathematics age PUMA Scale plus National Curriculum sublevel and APP level, as required. See Section 4: Obtaining and interpreting test s on page 18 for further details. 7

8 PUMA 3 Autumn answers and mark scheme Answer Category NC level 1 8, PS 1b 2 Join boy to the cylinder and girl to the triangular prism. geom 2b Both required = 3 1a = 9 Both required. No mark if more than these two ticked. 4 Any four small triangles shaded frac 2b 5 (a) 40 (b) 30 num num 2b 2a 6 27, 31 and 55 circled num 2c No mark if more than these three circled. 7 (a) 60 2c (b) 11 2b 8 (a) France stat 2c (b) Greece and Turkey stat 1a (c) 25 stat, PS 2c 9 14 children, PS 2a 10 (a) Pencil and Pad (b) A 20p, 5p and a 2p circled No mark for 10p, 10p, 5p, 2p. meas, PS meas, PS 11 14cm meas 12 A cross on the pentagon in the hexagons section geom 2a , 223, 230, 302, 320 num 2b 2b 2c 14 Any two identical numbers Accept two zeros and 103 num 2c 2c 445 and 145 num 2c 16 Crosses drawn on both parallelogram and right-angled geom 3a triangle only groups, PS c 19 Both 740 and 46 only num 20 Square correctly completed within 2mm of correct vertex, even if ruler not used geom 8

9 c 8 All three correct for 2 marks, two correct for 1 mark frac 3a 23 geom num, PS 3c a 26 (a) 40 minutes (b) any one section shaded on cake on left any two sections shaded on cake on right meas frac 3a 3a Both required Vertical faces need not be shaded frac frac 3a 4c frac 3a frac 3a 28 ¾ 1½ Both joined near enough to be unambiguous ½ ¾ 1½ a 30 30cm or 0.3m if unit altered meas, PS 4b PUMA 3 Autumn: Analysis of performance by category Category Number of National marks average mark National % Number Operations Fractions Measures Geometry Statistics Total Problem solving

10 PUMA 4 Autumn answers and mark scheme Answer Category NC level c 2 47p meas 2c 3 (a) D (b) C geom geom 2b 2a 4 (a) 25 children (b) 9 girls, PS 2c 2b 5 16mph meas 2a 6 Any two rectangles shaded frac 2a num 2b 8 24 and 48 num 2b Both required. 9 C and D only geom 3c cherries, PS 2a 11 num 3c rounds to 150 rounds to 160 rounds to All four numbers correct num 3c All three required. 13 (a) Isosceles triangle and trapezium ticked, only. geom 3c (b) Trapezium (correct spelling not required) geom 4b Do not accept quadrilateral minutes meas 3c 15 4 num 2a and 7 3 or 3 7 num 2a Both required. 17 (a) Ben and Sarah stat 2c (b) 15 (accept 15.00, but not 15.0) stat, PS 3a (c) 80 (accept 80.00, but not 80.0) stat, PS 4c frac Accept 7 3 / / 3 and 4 2 / 3 frac 3c Both required. 20 < < > 3a weeks meas 3a frac 4c 10

11 23 1 frac / c 1 / frac 4c 4 / jugs meas 3a eggs frac 4c 26 (8, 3) geom 4c num 3c Accept with or without space or comma or 938 num, PS 3a 29 (a) 9 flags, PS, PS 4b (b) 8 flags 4b b PUMA 4 Autumn: Analysis of performance by category Category Number of National marks average mark National % Number Operations Fractions Measures Geometry Statistics Total Problem solving

12 PUMA 5 Autumn answers and mark scheme Answer Category NC level 1 35 pebbles 2a 2 Any four rectangles shaded. frac 3 (a) 7 children (b) 22 children stat, PS stat, PS 2a 2a 4 2¾ and 1¼ frac 4b 5 Bottom right geom 3c mirror line 6 (a) 70 (b) Any one of 32, 33 or 34 num num 2a 3c 7 52 sweets, PS 3c 8 4.5cm 2 or 4½ cm 2 meas 4c and 2.1 Both required. frac 3c num 10 All four correct to to to to 800 over / 8 frac 12 (a) 3.54 (b) 6.46 Accept follow through, i.e. if the answer to (a) and (b) add up to 10 (c) 8 meas meas, PS meas, PS 3a 4c 12

13 a 14 (a) 764 (b) 647 or , 2, 3, 5, 6, 10, 15, 30 All required for the mark. Do not penalise incorrect ordering, the instruction is there to assist marking. 16 (a) 3 (b) (a) 6 edges (b) 8 faces (c) 12 vertices 18 (a) 20 (b) 75 (c) can t tell num, PS num, PS geom, PS geom, PS geom stat stat stat 2a 3c 4b 3a 4c 3a 4a 4a 3c 3a , PS 4a 20 (a) 60 (b) 90 (c) 40 frac frac frac 4a 5c num 3a 22 5 cups meas 23 1 hr 40 min meas 4a a 8 or 9 for 2 marks; 6 or 7 correct for 1 mark mark for both nearest 100 i.e and mark for both nearest 1000 i.e and (a) 100 (b) 7500 num num 4b 4a 4b 4b 13

14 27 2 frac / / 2 2 / frac 5c 28 (a) A and C only (b) C and E only geom geom 4b 5c and 29 only 4a 30 (a) 2 / 3 < 5 / 6 frac 4b (b) 3 / 4 > 8 frac 4b / a = 3cm b = 7cm Both required. meas 5c c PUMA 5 Autumn: Analysis of performance by category Category Number of National marks average mark National % Number Operations Fractions Measures Geometry Statistics Total Problem solving

15 PUMA 6 Autumn answers and mark scheme Answer and 1152 Both required. 2 4¾ and 5¼ respectively Both required. 3 (a) 22 (b) 90 Category num frac, SP frac NC level 3c 3c 3c 4 Rectangle and both triangles ticked only geom Accept 3.75p, 3-75, 3-75p. Do not accept 375 or 375p. 6 95km/h Accept meas meas 3c and 32 Both required. 15 and 45 Both required. 9, PS num 3c positioned closer to middle i.e. 2500, than 2000 or 3000 Accept arrows in the range 2250 and /4, 0.52 and 55% only frac 3a 11 number of edges number of faces number of vertices geom geom 4c 4a cube triangular prism square based pyramid All correct for 2 marks; any 2 rows correct for 1 mark. 12 Only 164g and 159g ticked num 4c 15

16 c 3a or 8 correct for 2 marks; 5 or 6 correct for 1 mark cm 15 (a) Range 85-86cm (b) Range months (c) Range 2-3cm (d) Range cm stat stat, PS stat, PS stat, PS 3a 4b 4b 6c weeks 4b 18 (a) 3 / 10 frac 4c (b) 1 / 3 frac 5b 19 kilograms 5kg 2.4kg ½kg grams 5000g 2400g 500g meas meas 12.5kg or 12½ kg g All correct for 2 marks; 2 correct for 1 mark. 20 (a) 128, PS 4c (b) 30 people 4a and num 4c 22 Tom frac, PS 4b b / 8, 6¼, 6½, 6 5 / 8, 6¾ frac 4a All required. 25 (a) 3000 (b) 60 num num 4c 3a 4c 26 1, 2, 3 and 6 only. Ignore the omission of 1 27 (a) 54cm 2 (b) 48cm meas meas 5b 5a 4a 16

17 28 Multiples of 7 are even Always Sometimes Never, PS 3a Prime numbers have exactly two factors, PS 6c Square numbers have an odd number of factors All correct for 2 marks; 2 correct for 1 mark frac 3a a All correct for 2 marks; 2 correct for 1 mark. frac 30 Top vertex geom, PS 5b frac 5c 32 28cm meas, PS 5b 33 (8, 10) geom, PS 5a 34 (a) (b) Accept answer for part (b) if it is greater than the answer for part (a) num, PS 6c 5c 5a PUMA 6 Autumn: Analysis of performance by category Category Number of National marks average mark National % Number Operations Fractions Measures Geometry Statistics Total Problem solving

18 4 Obtaining and interpreting test s Summative measures The results obtained from PUMA will enable you to report pupil performance in terms of: Age-standardised s (see Section 5) Mathematics ages (Table 4.2) National Curriculum sublevels (Table 4.3) APP level, subdivided as high, secure and low (Table 4.3) The PUMA Scale (Table 4.3). For a swift overview, you could compare how well a pupil has done by comparison to Table 4.1, which shows average s for each year group, by gender, for each PUMA test. You can also compare your own class average raw s against these averages. Table 4.1: Average test s by term and gender Autumn test Year Boys Girls Total

19 PUMA 3 Autumn raw Mathematics ages Many teachers use Mathematics ages as a quick reference: they shows the average chronological age of the pupils who obtained each particular raw (i.e. the chronological age at which this level of performance is typical). For more detailed comparative information, however, and especially for tracking progress over time, please refer to standardised s. Table 4.2: Mathematics ages for the Autumn term Mathematics age PUMA 4 Autumn raw Mathematics age PUMA 5 Autumn raw Mathematics age PUMA 6 Autumn raw Mathematics age 1 <7.0 1 <8:3 1 <9:0 1 <10:1 2 <7.0 2 <8:3 2 <9:0 2 <10:1 3 <7.0 3 <8:3 3 <9:0 3 <10:1 4 <7.0 4 <8:3 4 <9:0 4 <10:1 5 <7.0 5 <8:3 5 <9:0 5 <10:1 6 <7.0 6 <8:3 6 <9:0 6 <10:1 7 <7.0 7 <8:3 7 <9:0 7 <10:1 8 <7.0 8 <8:3 8 <9:0 8 <10:1 9 <7.0 9 <8:3 9 <9:0 9 <10:1 10 < <8:3 10 <9:0 10 <10:1 11 < <8:3 11 <9:0 11 <10:1 12 < <8:3 12 <9:0 12 <10:1 13 7:0 13 <8:3 13 <9:0 13 <10:1 14 7:2 14 <8:3 14 9:0 14 <10:1 15 7:4 15 8:3 15 9:1 15 <10:1 16 7:5 16 8:4 16 9:2 16 <10:1 17 7:7 17 8:7 17 9:4 17 <10:1 18 7:8 18 8:8 18 9:5 18 <10:1 19 7: :9 19 9:7 19 <10:1 20 8:0 20 8: :8 20 <10:1 21 8:1 21 9:1 21 9: :1 22 8:3 22 9:2 22 9: :3 23 8:4 23 9: : :5 24 8:6 24 9: : :8 25 8:7 25 9: : : :9 26 9: : :1 27 8: : : :4 28 9: : : :7 29 9: : : :9 30 >9:2 30 >10: : :0 31 >9:2 31 >10: : :2 32 >9:2 32 >10: : :5 33 >9:2 33 >10: :3 33 >12:5 34 >9:2 34 >10:2 34 >11:3 34 >12:5 35 >9:2 35 >10:2 35 >11:3 35 >12:5 36 >9:2 36 >10:2 36 >11:3 36 >12:5 37 >9:2 37 >10:2 37 >11:3 37 >12:5 38 >9:2 38 >10:2 38 >11:3 38 >12:5 39 >9:2 39 >10:2 39 >11:3 39 >12:5 40 >9:2 40 >10:2 40 >11:3 40 >12:5 41 >11:3 41 >12:5 42 >11:3 42 >12:5 43 >11:3 43 >12:5 44 >11:3 44 >12:5 45 >11:3 45 >12:5 46 >11:3 46 >12:5 47 >11:3 47 >12:5 48 >11:3 48 >12:5 49 >11:3 49 >12:5 50 >11:3 50 >12:5 19

20 Reporting progress the PUMA Scale In developing the PUMA tests, seven cohorts of 1,000+ pupils (just over 8,000 pupils in total) were tracked termly over four terms. Relating this data together statistically through the PUMA Scale enables you to link pupil performance from term to term and year to year. It provides a firm basis on which to project future performance. Table 4.3 provides a complete set of reference data for reporting progress for each test in terms of the PUMA Scale (and, for reference, National Curriculum and APP levels and LA points s). Find the raw in the column for the test your pupils have taken, then read across to obtain the level information in the form you require. Table 4.3: Relating s to different scales Years 3 4 PUMA Scale PUMA 3 Autumn raw PUMA 4 Autumn raw NC sublevels APP range Points PUMA Scale 1.0 <5 <3 1c 1L c 1L c 1S b 1S b 1S b 1S b 1S a 1S a 1H a 1H c 2L c 2L c 2S b 2S b 2S b 2S b 2S a 2S a 2H a 2H c 3L c 3L c 3S S S S S a 3S a 3H a 3H c 4L c 4L c 4S b 4S b 4S b 4S b 4S a 4S a 4H a 4H c 5L c 5L c 5S

21 Years 5 6 PUMA Scale PUMA 5 Autumn raw PUMA 6 Autumn raw NC sublevels APP range Points PUMA Scale 1.0 <3 1c 1L <3 1c 1L <3 1c 1S <3 1b 1S <3 1b 1S <3 1b 1S <3 1b 1S a 1S a 1H a 1H c 2L c 2L c 2S b 2S b 2S b 2S b 2S a 2S a 2H a 2H c 3L c 3L c 3S S S S S a 3S a 3H a 3H c 4L c 4L c 4S b 4S b 4S b 4S b 4S a 4S a 4H a 4H c 5L c 5L c 5S b 5S b 5S b 5S b 5S a 5S a 5H a 5H c 6L

22 Predicting future performance The tests for each term cover a range of demand appropriate to the year and term. In Table 4.4 below you can see at a glance the PUMA Scale of a pupil in any term and track to the next column to find the anticipated PUMA Scale they will obtain, if they make average progress. As the tests are designed to challenge pupils around the level at which they are expected to be working, you may find that pupils get similar raw s from term to term across a year, but their level of performance will continue to increase, as shown in the PUMA Scale,. You may wish to set targets, to provide an opportunity to measure the value added over a term or year. This is possible for both individual pupils and whole classes, by reference to the average performance data of over 1,000 pupils in each year group, from term to term and across all the years, in the standardisation sample. Table 4.4 provides this information. Monitoring the difference between the actual PUMA Scale and the predicted average PUMA Scale enables you to see if able children increasingly diverge from predicted progress or weaker children begin to converge toward normal progress. [See Table 4.4 on the next page] 22

23 Table 4.4: Monitoring and predicting progress from Autumn to Spring terms Average PUMA 3 Autumn PUMA Scale Average PUMA 3 Spring PUMA Scale Average PUMA 4 Autumn PUMA Scale Average PUMA 4 Spring PUMA Scale Average PUMA 5 Autumn PUMA Scale Average PUMA 5 Spring PUMA Scale Average PUMA 6 Autumn PUMA Scale Average PUMA 6 Spring PUMA Scale

24 Raw 5 Standardised conversion tables Age-standardised s for PUMA 3 Autumn Age in years and months 7:00 7:1 7:2 7:3 7:4 7:5 7:6 7:7 7:8 7:9 7:10 7:11 8:00 8:01 8:02 8:03 8:04 8:05 8:06 8:07 8:08 8:09 8:10 8:11 9:00 9:01 9: Award < 69 for all s in this area Award > 130 for all s in this area

25 Age-standardised s for PUMA 4 Autumn Raw Age In years and months 8:3 8:4 8:5 8:6 8:7 8:8 8:9 8:10 8:11 9:0 9: :3 9:4 9:5 9:6 9:7 9:8 9:9 9:10 9:11 10:0 10:1 10: Award < 69 for all s in this area Award > 131 for all s in this area

26 Age-standardised s for PUMA 5 Autumn Raw Age in years and months 9:0 9:1 9:2 9:3 9:4 9:5 9:6 9:7 9:8 9:9 9:10 9:11 10:0 10:1 10:2 10:3 10:4 10:5 10:6 10:7 10:8 10:9 10:10 10:11 11:0 11:01 11:2 11: Award < 69 for all s in this area [Table continues on next page] 26

27 [PUMA 5 Table cont.] Raw Age in years and months 9:0 9:1 9:2 9:3 9:4 9:5 9:6 9:7 9:8 9:9 9:10 9:11 10:0 10:1 10:2 10:3 10:4 10:5 10:6 10:7 10:8 10:9 10:10 10:11 11:0 11:01 11:2 11: Award > 131 for all s in this area

28 Age-standardised s for PUMA 6 Autumn Raw Age in years and months 10:1 10:2 10:3 10:4 10:5 10:6 10:7 10:8 10:9 10:10 10:11 11:0 11:1 11:2 11:3 11:4 11:5 11:6 11:7 11:8 11:9 11:10 11:11 12:0 12:1 12:2 Award < 70 for all s in this area [Table continues on next page] 28

29 [PUMA 6 Table cont.] Raw Age in years and months 10:1 10:2 10:3 10:4 10:5 10:6 10:7 10:8 10:9 10:10 10:11 11:0 11:1 11:2 11:3 11:4 11:5 11:6 11:7 11:8 11:9 11:10 11:11 12:0 12:1 12: Award > 131 for all s in this area

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