Critical_Thinking_and_Problem_Solving_Th

Another way of approaching problems is to In this diagram, the outer box represents all lay out the information in a different way. This the students (the universal set); in this case is especially so when the information is given they all take physics. The left-hand circle verbally – and therefore the connection represents those taking chemistry (and between the different pieces may not be physics) and the right-hand circle represents immediately obvious. Consider, for example, those taking biology (and physics). the following problem: We know that the number taking only Activity physics is 12; this is represented by the area outside both circles. Those taking all three In a group of 45 students at a school, all sciences are represented by the intersection students must study at least one science. of the two circles and shown as 9. The Physics is compulsory, but students may also number taking both physics and biology is opt to study chemistry or biology or both. 9 24; of these 9 take all three, so 15 take only students take all three sciences. 24 take physics and biology. This is shown by the both physics and biology (with or without outer section of the right-hand circle. We can chemistry). 12 students take only physics. now calculate the number in the area marked by the question mark, as this must be all the How many are studying chemistry and students in the class minus the numbers in physics but not biology? the other three areas, i.e. 45 − 12 − 9 − 15 = 9. This is the required answer: the number Commentary studying chemistry and physics but not With a bit of clear thinking, this may be solved biology is 9. in a direct fashion by making an intermediate calculation (those not studying biology). Since Interestingly, the number studying all three all students take physics, the situation is was not used in the original calculation. It is, simplified. 24 study biology and there are 45 in fact, not needed to solve the problem. We in total, so 21 do not study biology. These 21 used it in the Venn diagram solution so we comprise those studying physics alone and could calculate the numbers in all the areas on those studying both physics and chemistry. the diagram. However, we know that 12 take only physics, so 9 must take physics and chemistry but not Graphs, pictures and diagrams can often be biology. useful in solving problems as they help to clarify the situation and represent the Although this appeared to be an easy numbers used in a more digestible manner. calculation, the method of approach was not This is covered in more depth in Chapter 6.2. obvious. The situation can be made a lot easier by using a Venn diagram: The activity above shows that problems can often be solved in more than one way. It is Physics important to keep the mind open to 12 Chemistry alternatives and not always to pursue a method which is not apparently leading to a solution. Biology ? 9 15 3.5 Finding methods of solution 95

Activity questions – that is, checking that we have the right answer. There is one railway on the island of Mornia, which runs from Enderby to Widmouth. There If ER takes 24 minutes (including a are two intermediate stops at Maintown and 3-minute stop) and MR takes 18 minutes (no Riverford. The trains run continuously from stop), then EM (no stop) must take 3 minutes one end to the other at a constant speed, (24 − 18 − 3). Similarly, RW must take 36 − 18 stopping for three minutes at each station. − 3 = 15 minutes. We now have the times for From departing Enderby to arriving at all the sections: Widmouth takes 42 minutes. From Enderby to Riverford takes 24 minutes. From Maintown EM = 3 minutes to Widmouth takes 36 minutes. Stop at M = 3 minutes MR = 18 minutes How long does it take from Maintown to Stop at R = 3 minutes Riverford? RW = 15 minutes Commentary The total of all these is 3 + 3 + 18 + 3 + 15 A diagram makes this problem much easier to = 42 minutes as expected. solve. Check that the times from E to R and M to 42 min W agree with the answer of 18 minutes from M to R as given. E MR W 24 min This example once again shows that representing the data in a different way can 36 min lead to a simple method of solving a problem that at first appears unclear. Summary It can now be seen that if we add the time from • We have learned the importance of finding E to R to the time from M to W, we get the time methods of solution for problems for which from E to W plus the time from M to R. The the way of proceeding to an answer is not times from E to R and M to W each include one necessarily obvious. 3-minute stop, whilst the time from E to W includes two 3-minute stops, so when we • We have found that looking for subtract the EW time from the sum of the ER intermediate results may help to lead to and MW times, the stops cancel out. Thus the the final answer. time from M to R is 24 + 36 − 42 = 18 minutes. • We also looked at the value of alternative We can now look further at an extra useful ways of presenting data and considering element in the solution of problem-solving more than one way of solving problems. 96 Unit 3 Problem solving: basic skills

End-of-chapter assignments 3 From my holiday cottage by the sea I can see two lighthouses. The southern flashes 1 Aruna’s neck chain has broken into two regularly every 11 seconds. The northern parts. She has lost the broken link and lighthouse, after its first flash, flashes is having it repaired by a jeweller who will again after 3 and 7 seconds. The whole open one of the remaining links and use it cycle repeats every 17 seconds. to rejoin the chain. The chain is made from metal 2 mm thick and each of the broken  They have just flashed at exactly the pieces has a fitting at the end used for same time, the northern one having just closing the chain which each adds 1 cm to started the cycle described above. the total length.  When will they both flash again at  exactly the same time? One of the broken pieces is 34.2 cm long 4 The 23 members of a reception class in a and has 10 more links than the other, school have done a survey of which cuddly which is 26.2 cm long. Excluding the toys they own. Pandas and dogs are the fittings at the ends, how many links will most popular, but 5 children have neither there be in the complete chain? a panda nor a dog. 12 have a panda and 2 T he distance from Los Angeles to Mumbai 13 have a dog. How many have both a dog is 14,000 km. Flights take 22 hours, and a panda? whilst the return flight from Mumbai to Los Angeles takes only 17 hours because Answers and comments are on pages 317–18. of the direction of the prevailing wind. Assuming the aeroplane would fly the same speed in both directions in still air, what is the average wind velocity? 3.5 Finding methods of solution 97

3.6 Solving problems by searching Some problems may not always be resolved by We can then continue with the first coin as a using direct methods of calculation. 5¢ in the same manner (we do not need to Sometimes, problems do not have a single consider repeats). The possibilities are: solution, but many, and we need to find one that represents a maximum or minimum (for 1 ¢ + 1¢, 1¢ + 5¢, 1¢ + 10¢, 1¢ + 25¢, 1¢ + example the least cost or shortest time for a 50¢, then journey). In these cases we need to have a 5¢ + 5¢, 5¢ + 10¢, 5¢ + 25¢, 5¢ + 50¢, systematic method of evaluating the data to 10¢ + 10¢, 10¢ + 25¢, 10¢ + 50¢, come up with all (or at least the most likely) 25¢ + 25¢, 25¢ + 50¢, and possibilities. This is called a ‘search’. Once 50¢ + 50¢. again, with this type of question, it is important to have a way of checking that the Listing all the totals, we have: 2¢, 6¢, 11¢, 26¢, final answer is correct. 51¢, 10¢, 15¢, 30¢, 55¢, 20¢, 35¢, 60¢, 50¢, 75¢ and $1. Here is an example of a problem that requires a search. Finally, we need to list all the possibilities with three coins. This is slightly more difficult. Activity However, we only need to go on until we have found an impossible amount (you may already Amir is helping with a charity collection and have spotted it). The possibilities are: has gathered envelopes containing coins from a number of donors. He notes that all 1¢ + 1¢ + 1¢, 1¢ + 1¢ + 5¢ etc. the envelopes contain exactly three items but 1¢ + 5¢ + 5¢ etc. some of them contain one, two or three buttons instead of coins. All the coins have You should have spotted by now that we have denominations of 1¢, 5¢, 10¢, 25¢ or 50¢. not seen the value 4¢ and that all further sums of three coins (anything including a 5¢ or above) What is the smallest amount of money will be more than 4¢. So 4¢ is the answer. that is not possible in one of the envelopes? This was actually a trivial example used for Commentary the purposes of illustration. There is an The easiest way to approach this question is to alternative way to solve this, which also list the possibilities in a systematic order. We involves a search. This is to look at 1¢, 2¢, 3¢, know envelopes can contain 0, 1, 2 or 3 coins. etc. and see whether we can make the amount The possibilities with one coin (and two up from one, two or three coins. In this case it buttons) are: 1¢, 5¢, 10¢, 25¢ or 50¢. would have led to a very fast solution, but if the first impossible value had been, for That was the easy part. With two coins (and example, 41¢, this second method would have one button), we need to be a little more taken a very long time and we might have careful. First consider that the first coin is 1¢, been unsure that we checked every possible then look at all the possibilities for the second. sum carefully. The method described above is called an ‘exhaustive search’, where every possible 98 Unit 3 Problem solving: basic skills

combination is considered. The alternative method involves analysing the problem, given in the above paragraph may be called a which can be a very useful tool in reducing the ‘directed search’ where we are looking size of searches. selectively for a solution and will give up the search once we have found one. In the case The type of search shown above involves above, where we were looking for a minimum, combining items in a systematic manner. we can reasonably start searching from the Other searches can involve route maps – lowest value up. looking for the route that takes the shortest time or covers the shortest distance, or A third alternative may be described as a tables – for example finding the least ‘selective search’. In this case we are using a expensive way of posting a number of parcels. partial analysis of the problem to reduce the size of the search, concentrate on certain With all these searches, the important areas, or to reject unlikely areas. The activity thing is to be systematic in carrying out the below illustrates this. search so that no possibilities are missed and the method leads to the goal. The activity Activity below involves finding the shortest route for a journey. Try repeating this exercise using coins of denominations 1¢, 2¢, 5¢, 10¢, 20¢ and Activity 50¢ and with 1 to 4 coins in each envelope. This is quite a long search. Consider (and The map shows the roads between four discuss with others) whether there are ways towns with distances in km. of shortening it. I work in Picton and have to deliver Commentary groceries to the other three towns in any If you start this search you will find it takes a order, finally returning to Picton. What is the very long time. It is difficult to be absolutely minimum distance I have to drive? systematic (especially when considering all options for four coins). It is also difficult to 14 keep track of all values that have been covered at any point in the search. It is necessary to Queenstown Picton look for short-cuts, and out of boredom you will probably have done so. 10 Roseford 22 18 12 The denominations 1¢, 2¢ and 5¢ in combinations of 1 to 3 coins can make all the 8 values from 1¢ to 10¢. This means that, by adding to the 10¢ and 20¢ coins, all amounts 5 from 1¢ to 30¢ can be made. After 30¢ it is necessary to use both the 10¢ and 20¢ or a 20¢ Southland and 2 × 5¢. The former leaves one or two extra coins, which can make 1¢, 2¢, 3¢, 4¢, 5¢, 6¢, Commentary 7¢ but not 8¢. The latter leaves only 1 coin, There are only a small number of possible which cannot be 8¢, so 38¢ is the minimum routes. If these are laid out systematically and that cannot be made from 1 to 4 coins. This the sums calculated correctly, the problem can be solved quite quickly. The possible routes (with no repeated visits to any towns – you should be able to satisfy 3.6 Solving problems by searching 99

yourself that it is never worth retracing your Therefore, the minimum distance is 59 km. steps) are: If you were particularly astute, you PQRSP would have noticed that the routes come in PQSRP three pairs of the same distance (e.g. PQRSP PRQSP is the reverse of PSRQP so must be the same). PRSQP This would have saved you half the PSRQP calculations. PSQRP Summary This gives six values. In order to see how they are obtained you may note that there are three • We have learned that some problems pairs, each pair visiting a different one of the require a search to produce a solution. towns first (after leaving Picton). The two routes in each pair take the last two towns in • We have seen the importance of being opposite orders. systematic with a search, in order both to ensure that the correct answer is obtained The distances associated with each route and to be certain that we have the right are as follows: answer. 59 km • We also saw that searches do not always 62 km have to be exhaustive and how analysis 67 km of the problem can reduce the size of the 62 km search and time taken. 59 km 67 km End-of-chapter assignments 1 The notice below shows admission prices Additional adult $10 to the Tooney Tracks theme park. Adult $12 Family ticket (for 1 adult and 2 $20 children) Child (aged 4–16) $6 Additional child 4–16 or senior $5 citizen Child (aged under 4) Free Senior citizen $8 Additional adult $10 Family ticket (for 2 adults and $30 Maria is taking her three children aged 2 children) 3, 7 and 10 and two friends of the older children (of the same ages) as well as her Additional child 4–16 or senior $5 mother, who is a pensioner. What is the citizen least it will cost them? 100 Unit 3 Problem solving: basic skills

2 I recently received a catalogue from a book  If 5¢ and 20¢ coins are the same club. I want to order seven books from thickness, how many different heights of their list. However, I noticed that their price $1 pile could she have? structure for postage was very strange: A 5  B 6  C 10  D 11  E 20 Number of Cost of post 4 In a community centre quiz evening, teams items and packing were awarded five points for a correct answer, no points for no answer, and minus 1 45¢ two points for an incorrect answer. The teams marked their own score sheets. I 2 65¢ arrived late and the scores after seven questions were shown on the board as 3 90¢ follows: 4 $1.20 Happy Hunters 28 5 $1.50 Ignorant Idlers 18 6 or more $3.20 Jumping Jacks 16 I decide, on the basis of this, that I will ask Kool Kats 12 them to pack my order in the number of parcels that will attract the lowest post and Lazy Lurkers -1 packing charge. How much will I have to pay? a One team was clearly not even clever 3 Jasmine has been saving all year for her enough to calculate their score correctly. brother’s birthday. She has collected all Which one was it? the 5¢ and 20¢ coins she had from her change in her piggy bank. She is now b Are there any scores, other than those counting the money by putting it into piles, shown above, that would have raised all containing $1 worth of coins. She suspicion? notices that she has a number of piles of different heights. Answers and comments are on pages 318–19. 3.6 Solving problems by searching 101

3.7 Recognising patterns In Chapter 3.1 we saw that there are three An extension of this skill is to identify main skills involved in solving problems. We possible reasons for variation in data – once have already dealt with the first two of these again, this springs from past experience as to (identifying important information and what causes changes and the types of combining pieces of information). This variation that may be expected. This type of chapter deals with the third skill, that of question is dealt with in more detail in identifying pieces or sets of information in Chapter 3.8. different forms which are equivalent. In particular, this chapter deals with graphical, These are best illustrated using examples. verbal and tabular information. The first deals with identifying the similarity between two sets of data. Activity The table shows the results of a survey into Which of the bar charts accurately ownership of various household appliances represents the data shown below? by families who live in a town. Appliance Dishwasher Vacuum Washing Microwave Food processor Toaster cleaner machine oven 34 92 % ownership 68 98 77 54 A B C 100 100 100 % ownership % ownership % ownership 50 50 50 Dishwasher Toaster Vacuum cleaner Food processor Washing machine Microwave oven Microwave oven Washing machine Food processor Vacuum cleaner Dishwasher Toaster 000 Vacuum cleaner Toaster Washing machine Dishwasher Microwave oven Food processor % ownershipD Vacuum cleaner E 100 Toaster 100 50 50 Vacuum cleaner 0 Washing machine 0 Toaster Dishwasher Washing machine Microwave oven Dishwasher Food processor Microwave oven % ownership Food processor 102 Unit 3 Problem solving: basic skills

Commentary She drew one pie chart last night, but has This question is actually quite easy. It is only a not labelled the segments and cannot matter of being careful and matching remember which country it represents. The appliance to bar length correctly. The main pie chart is shown below. complication (and this is a potential trap for those who don’t look at the question and the Which country is it? graphs carefully) is that the order of the appliances in the graphs is different in some Commentary cases from the order in the table. Also, the The data expressed in local currency is not very exact heights of the bars cannot always be read useful for a direct comparison. It is easier if the accurately enough at the scale on which the costs are all expressed as percentages so that graphs are drawn, so it is necessary to look at the appropriate components can be compared. the relative heights of the different bars. The table is repeated below with the costs as percentages of the totals for each country. In fact D is the correct graph. The appliances have been put into order by their percentage Sudaria Idani Anguda Boralia ownership. A has the appliances ordered as for D but the bars are in the order of the table. The Crude oil 51.09 36.00 30.10 47.62 other graphs have similar errors – you might like to identify the error in each case. Refining 1.46 9.00 1.50 1.90 Activity Wholesale 6.57 7.00 3.76 13.33 This activity reverses the skill shown above: the graph is given (a pie chart in this case) and the cost structure it represents has to be identified. A student is drawing pie charts to show how the various elements of the cost of fuel contribute to the total price in various countries. The data she is using is shown below, with the prices in local currencies. Sudaria Idani Anguda Boralia Retail 4.38 4.00 4.36 4.76 Crude oil 0.70 18.68 0.40 0.50 Tax 36.50 44.00 60.27 32.38 Refining 0.02 4.67 0.02 0.02 In the pie chart, the largest segment is just under half the area. It could, therefore, only be Wholesale 0.09 3.63 0.05 0.14 tax in Idani or crude oil in Boralia. We cannot easily distinguish which as the difference in Retail 0.06 2.08 0.06 0.05 the second-largest segments is not very great. We must, therefore, look at the smallest three Tax 0.50 22.84 0.80 0.34 segments. Boralia has one (wholesale) three times the size of either of the other two, but in Total 1.37 51.90 1.33 1.05 the pie chart they are much closer than this, so the answer must be Idani. 3.7 Recognising patterns 103

This could, in fact, have been solved without Commentary going to percentages by looking at the relative This question is a little more difficult than sizes of the components in the table for each some we have seen so far. There are several country. It would have been quicker to do this, ways to approach it. We can note that if we but would have taken more mental arithmetic. knew the actual averages for the four colleges the newspaper did include, it might be possible Activity to see if these averages disagreed with an estimated average for the five colleges, and the The table shows the results of a questionnaire, direction of the error would give some asking the five colleges in a town the proportion indication of which one was forgotten. of students taking 1–4 A Level subjects. Looking at the ‘averages’, the approximate Percentage of students taking values (we have to estimate these from the number of A Levels shown graph) are 9, 19, 35 and 16 respectively for 1, 2, 3 and 4 A Levels. Multiplying these by 54 (to College 1 2 3 4 correct for the fact that they were divided by 5 instead of being divided by 4), we get Abbey Road 13 25 42 20 (approximately): 11, 24, 44 and 20. Barnfield 5 18 55 22 If we were being very systematic, we could now compare these with all sets of four Colegate 24 36 28 12 averages, but it would take a long time. Instead, let us note that the 11 looks a little low Danbridge 16 18 61 5 for the average of 1 A Level, as does 24 for the average of 2. 44 for the average of 3 looks very Eden House 10 14 48 28 low and 20 for the average of 4 looks far too high. From this, we may suspect that The local newspaper (forgetting that there might Danbridge has been missed as it is higher than be different total numbers of students in the the others for 3 A Levels and lower for 4. five colleges) just added the numbers together and divided by five to produce a percentage We can check this by averaging one of the graph for the town as a whole. However, they columns for the other four colleges (preferably forgot to add in the data for one college so their use 3 or 4 A Levels as they look to have the percentages did not add up to 100. biggest discrepancy) and comparing the results – try this for yourself and see whether you can confirm that Danbridge is the college whose results are missing. Percentage of students30 Summary 20 • We have learned how data may be represented in more than one way and the 10 importance of systematic comparisons between two sets of data in ascertaining 0 that they are the same. 1234 Number of A Levels • We saw that reading graphs and tables carefully is necessary in order not to make Which one did they forget? errors in identifying similarities. 104 Unit 3 Problem solving: basic skills

End-of-chapter assignments 1 Look in newspapers (business pages are played? (Hint: you first need to decide often useful) or on the internet to find which games have already been played, so examples of numerical data in various forms you know what is left.) (verbal, graphical, tabular). Express the data in a different form. Consider which form ABCDE makes the data clearest to understand. Britons 5 3 5 5 2 2 Four-digit personal identification numbers (PINs) are used to withdraw cash from Danes 54755 banks’ machines using plastic cards. It can be very difficult to remember your Normans 3 4 3 4 6 personal number. I have a method of remembering mine. It is the two numbers Saxons 2 4 1 1 2 of my birth date (i.e. the date in the month) reversed, followed by the two digits of my 4 The graph shows the charges made byTotal cost ($) month of birth reversed (using a zero in a printing company for making various front if either is a single number so, for numbers of posters. example, May would be 05). 250  Which of the following could not be 200 my PIN? 150 A 3221   B 5060   C 1141   D 2121   E 1290 100 3 Four house teams play each other in a 50 school basketball league. The scoring system gives three points for a win, one for 0 a draw and none for losing. 1234567 Number of posters  They all play each other once, and the league table before the last round of Which of the following pricing structures matches is as follows: would give the graph shown? Played Won Drawn Lost Points A $30 per poster B $50 set-up charge + $20 per poster Britons 2 0202 C $40 per poster for the first four, any Danes 2 1104 extra $20 each D $30 set-up charge, $30 per poster Normans 2 1 0 1 3 for the first four, any extra $20 each Saxons 2 0 1 1 1 Draw the graphs for the other price structures. Answers and comments are on page 319. Which of the following points columns are possible after the last two matches are 3.7 Recognising patterns 105

3.8 Hypotheses, reasons, explanations and inference In the introductory chapters we saw thatYeast Which of the following explanations are problems involving making inferences fromconcentrationconsistent with the shape of the curve? data or suggesting reasons for the nature of (Identify as many as apply.) variations in the data may appear in either the critical thinking or the problem-solving A Yeast cells divide when they have sections of thinking skills examinations. grown enough, so grow exponentially if they have enough nutrient. Such examples are usually based on quantitative (numerical or graphical) data and B The rate of increase of yeast cells may arise from such areas as finance or depends only on the amount of science. They require analysis of the data given nutrient. in order to reach some conclusions that may be drawn from the data or to suggest reasons C Eventually, the growth of yeast cells is for the nature of the data. limited by lack of nutrient. The example below is based on a scientific D Yeast cells die when there is scenario. While this requires a little insufficient nutrient. understanding of basic scientific concepts, most of the skills involved in coming to a E The shape of the curve is explained by solution depend on clear, logical thinking. a linear growth in yeast and a linear decrease in nutrient. Activity Commentary The graph shows the results of an experiment Looking at the statements in turn: to determine the growth of a culture of yeast in a nutrient medium. The liquid containing A This statement explains the initial the nutrient was made up and a small amount increase in growth rate – the increase of yeast introduced. At regular intervals looks exponential (increasing in size at a afterwards, the solution was stirred, a small constantly growing rate). sample taken and the concentration of yeast measured. The graph represents a smooth B This statement would not explain the line drawn through the results. initial growth – it would start at a higher growth rate, which would then decrease Time all the time. C This statement would explain the drop to zero growth after a time, linked to a lack of nutrient. D There is no indication of death; in that case the population would fall. E If both processes were linear (resulting in straight-line graphs), a combination of them would also result in a straight-line graph. 106 Unit 3 Problem solving: basic skills

So A and C are the explanations for the shape Commentary of the curve. Nikul arrives at the station at either 15 or 45 past the hour. Therefore, he takes the train The activity below is firmly in the either on the hour or at 20 past the hour. He Cambridge Thinking Skills syllabus category of gets to the bus stop at 20 or 40 past the hour. ‘suggesting hypotheses for variations’. You are Buses at 5, 25 and 45 past the hour would given a scenario incorporating numerical data, therefore fit the requirements: and asked which, of a number of possible situations, could explain the nature of the data. • H e would get the bus at 25 past the hour if he arrived at 20 past. This would mean Activity that he arrived at work 5 minutes late. Nikul runs exercise classes at his local gym, • H e would get the bus at 45 past the hour and gets there each day by train and bus. if he arrived at 40 past. This would mean Classes start at different times each day, but that he arrived at work 5 minutes early. always either on the hour or at half past the hour. He always gets to the railway station • H e would never use the bus at 5 past the 45 minutes before he is due to start teaching hour, so it doesn’t matter that this one and the train journey takes 20 minutes, after doesn’t fit the arrival times. which he takes a bus to the gym, which takes 10 minutes. Trains leave every 20 minutes, The correct answer is D. It is also illustrative to starting on the hour. Some days Nikul finds see why the wrong answers do not work: that he gets to work 5 minutes early. On all the other days he finds that he gets there A Buses at 5 and 35 past the hour would 5 minutes late. always get Nikul to work on a quarter hour which could not be 5 minutes early Which one of the following could explain or late. the times that Nikul arrives at the gym? B Buses at 15 and 45 past the hour would A The buses leave at 5 and 35 past mean that Nikul was always 5 minutes each hour. early. B The buses leave at 15 and 45 past C Buses at 25 and 55 past the hour would each hour. mean that Nikul was always 5 minutes late. C The buses leave at 25 and 55 past E Because Nikul arrives on the train at 20 or each hour. 40 past the hour, he would be getting the 35 or 55 past the hour bus. The bus at 35 D The buses leave at 5, 25 and 45 past past the hour would get Nikul to work at each hour. 15 or 45 past the hour which is neither 5 minutes early nor 5 minutes late. E The buses leave at 15, 35 and 55 past each hour. Longer questions at A Level can involve analysing quite complex data and determining what conclusions may be drawn from it. The activity below is of this type. It looks at identifying reasons for variations in data. 3.8 Hypotheses, reasons, explanations and inference 107

ActivityWembling in ation as a percentage B A rise in Danotian inflation would only of Danotian in ation cause the fall in the ratio shown if the rise The graph shows the inflation rate in the in inflation in Wembling was smaller or province of Wembling as a percentage of negative: we know nothing about this. inflation in the country of Danotia as a whole over the period from 2006 to 2012. C If food prices rose less in Wembling than in Danotia as a whole, this could explain 100 why the ratio fell – even if inflation in Wembling was rising. 90 D Seasonal fluctuations would only manifest 80 themselves within a year, not between years. 70 2006 2007 2008 2009 2010 2011 2012 E Even if the inflation rate in Wembling is falling, we know nothing about the inflation Which of the following is the most plausible rate in the whole of Danotia, so we cannot explanation for the variations shown in the conclude that the ratio would fall. graph? Thus C is the only reasonable answer. The A Danotian inflation has been high over others depend either on reading the graph the period shown. incorrectly or reading more into the graph than we can safely conclude. This illustrates B Danotian inflation has risen over the the importance of reading and understanding period shown. the information given (both verbal and in other forms) and of reading the question C Food prices in Wembling have risen correctly. Beyond that, the deductions that less than in Danotia as a whole. can be made follow from the application of correct logic. D Food prices in Wembling are subject to higher seasonal fluctuations than in As in the previous chapter, solving these Danotia as a whole. two types of problem also depends on the skill of recognising an identity between data E The inflation rate in Wembling is falling presented in two forms. As with the other due to high unemployment. skills described here, this comes with practice and it can be useful to look at data in Commentary newspapers to see how they are presented Again we will look at these five answers in and to consider whether they are always turn. It is important to remember that the presented in the clearest way. graph represents the inflation in Wembling as a percentage of the inflation in Danotia, not The next activity uses logic based more on the actual inflation rate in Wembling. manipulating numbers. In this regard, it has elements of the ‘finding methods of solution’ A The graph represents only the ratio skill from Chapter 3.5. However, the nature of between Wembling and the whole of the question places it closer to the ‘suggesting Danotia; high inflation in Danotia cannot hypotheses for variations’ category of questions. explain the shape of a graph of the ratio. 108 Unit 3 Problem solving: basic skills

Activity they leave Whitesea at 9.00 a.m., 9.20 a.m., 9.40 a.m., etc. They then leave Greylake 1 hour (Harder task) A tram company runs a service 10 minutes later, at 10.10 a.m., 10.30 a.m., along the seafront from Whitesea to Greylake. 10.50 a.m., etc. A driver leaving Whitesea, for Trams leave Whitesea at regular intervals, example, at 11.20 a.m. will see the trams which starting from 9 a.m., taking one hour to reach left Greylake at 10.30 a.m., 10.50 a.m., Greylake. They then turn around and start 11.10 a.m., 11.30 a.m., 11.50 a.m. and back 10 minutes after arriving. A driver in the 12.10 p.m. – six in total. middle of the day, in his journey from Whitesea to Greylake, always sees six trams travelling in Looking at statement A, the first tram the opposite direction. Some of these will leaving Whitesea at 9.00 a.m. reaches Greylake have set off from Greylake before he left, and at 10.00, leaves at 10.10 and returns to some will have set off after he left. Whitesea at 11.10, in time to become the 11.20 service. In the meantime, other trams will Which of the following must be true? There have left at 9.20, 9.40, 10.00, 10.20, 10.40 and may be more than one. If any of the 11.00, i.e. six more, so there must be seven statements are not true, can you correct trams to run the service. A is incorrect. them? Looking at statement C, if I sit near the A It takes six trams to run the service. midpoint from 11.15 a.m. to 12.15 p.m., I will see B The trams run every ten minutes. the 11.00, 11.20 and 11.40 trams going one way C If I sit on the seafront from 11.15 a.m. and the 10.50, 11.10 and 11.30 trams going the other way, so I will see six in total. C is correct. to 12.15 p.m., I will see six trams going past. Can you confirm these answers by constructing a timetable? Commentary We can test the statements by making a This kind of problem will be revisited in timetable. However, to do this, we need to make Chapter 6.2 where we look at graphical an assumption about the departure intervals. It solutions to problems. is, in fact, better to carry out a little analysis first. Summary When a tram driver leaves Whitesea, all those trams which left Greylake up to one • We have encountered examples where we hour earlier will already be on their journey. are required to suggest a hypothesis or a During the hour it takes him to travel to reason for the nature of variation in data. Greylake, more trams will be leaving Greylake. So, in total, he will see all those trams which • The terms hypothesis, reason, explanation left up to an hour before he left, and also all and inference are used in exactly the those which leave up to an hour after he left. same way in problem solving as in critical This means the six trams he sees must have left thinking, the only difference being that the Greylake in a two-hour period. As they leave at information given is in the form of data regular intervals, one must leave every rather than verbal description. 20 minutes, so B is incorrect. • We have seen that extended examples This may be illustrated by looking at some where more data is supplied can require actual times. If trams leave every 20 minutes, analysis that may lead to a range of conclusions. 3.8 Hypotheses, reasons, explanations and inference 109

Average e ectivenessEnd-of-chapter assignments at half the time shown in the graphs, patients have stopped taking it. Average e ectiveness1 In order to treat a particular disease  Assuming that the effects of the effectively, patients are initially given two two drugs are independent, what would drugs. Drug A alone has the effect shown be the expected shape of the graph of on the graph below. (10 = total relief from effectiveness for a patient on the regime symptoms. 1 = no relief.) described? 10 Drug A 2 A t a local school, 70% of the students studied French and 45% studied German. 1 Which of the following can be confirmed Time from the information given? The effect of drug B showing how it varies A All students study either French or with time is shown in the graph below (on German. the same effectiveness and time scales). 10 Drug B B 23 of those studying German do not study French. 1 Time C 25% of the students study neither French nor German. The reason the patients are given two drugs is that drug A, whilst being very D At least 15% of students study both effective, has long-term harmful side French and German. effects. Drug B takes some time to become effective, and has a lower eventual 3 I was shopping at a market in Northern effect but can be taken indefinitely. The Bolandia and asked a local how much an regime used by doctors is to give both orange was. He said that an orange and a drugs starting at the same time, then to lemon together cost $2. He then further withdraw A at a uniform steady rate until, confused me by saying that a grapefruit and a lemon cost $3 and that all three were different prices.  Based on this rather unhelpful information, which one of the following can be confirmed? A An orange costs more than a lemon. B A lemon costs more than a grapefruit. C A grapefruit costs more than a dollar. D An orange costs less than a dollar. 110 Unit 3 Problem solving: basic skills

4 The Fitland health centre swimming pool Which of the following is possible? is open seven days a week. There are four lifeguards, Liam, Moses, Nadila and A Liam works on Thursday. Orla, each working four days a week. None B Nadila works on Sunday and works four consecutive days and at least two lifeguards are on duty each day, with Monday. three on duty on Saturday and Sunday. C Orla works on Monday and Tuesday. We know that: D Three people work on a Monday. • Liam doesn’t work Monday or Saturday. • Moses doesn’t work Tuesday, Thursday Answers and comments are on pages 319–21. or Friday. • Nadila doesn’t work Wednesday or Friday. • Orla always works on Thursday. 3.8 Hypotheses, reasons, explanations and inference 111

3.9 Spatial reasoning Spatial reasoning involves the use of skills that Approximately what proportion of the two are common in the normal lives of people tiles will be needed to cover the whole floor? working in skilled craft areas. Imagine, for example, the skill used by joiners in cutting roof A  three white to one black joists for an L-shaped building. This is also B  two white to one black necessary for many professionals: the surgeon C  equal quantities of both needs to be able to visualise the inside of the D  two black to one white body in three dimensions and, of course, E  three black to one white architects use these skills every day of their lives. Commentary Spatial reasoning questions can involve This seems a relatively simple problem, but the either two- or three-dimensional tasks, or answer is not immediately obvious. This is an relating solid objects to flat drawings. example of a tessellation problem. There are Thinking in three dimensions is not various ways to go about solving it – one way is something that comes easily to all people, but to continue drawing the pattern until you have undoubtedly practice can improve this ability. enough tiles that you can estimate how many of each are needed. Another, more rigorous, In the simplest sense, a problem-solving method is to identify a unit cell that consists of question involving spatial reasoning can a number of each tile, which may be repeated require visualising how an object will look as a block to cover the whole area. Such a unit upside down or in reflection. More cell for this problem is shown in the drawing. complicated questions might involve relating a three-dimensional drawing of a building to a view from a particular direction or the visualisation of how movement will affect the view of an object. This chapter is shorter in terms of description than most of the others but there are more examples at the end; this is an area where practice is more important than theory. The next example involves a problem- solving task in two dimensions. Activity If you now think carefully, you can imagine that this block of three tiles could be repeated The drawing shows part of the tiling pattern used for a large floor area in a village hall. This is made up of two tiles, one circular (shown in black) and one irregular six-sided tile (shown in white). 112 Unit 3 Problem solving: basic skills

over and over again, filling any area without Commentary any gaps, to give the pattern shown in the There must be some sort of recess in the original drawing. So two white tiles are needed top-left corner (top back right as shown on the for every black tile. B is the correct answer. 3-D drawing). We cannot tell whether it goes right to the bottom as shown in A and B or just The activity below involves three- some of the way down as shown in C, D and E. dimensional reasoning. Because the drawing is not of a familiar object, there are no short- Similarly, there is a recess that goes through cuts; you need to work out what the to the right-hand edge (left back on the 3-D possibilities are for the unseen side. drawing). Those shown in A, C and E would all give the same 3-D view from the front. We can Activity now eliminate A, C and E, as both the rear features are shown as being possible. The drawing is a three-dimensional (Remember that we are looking for the representation of a puzzle piece. diagram that is not a possible representation.) D joins up the two recesses – this would also be Which diagram is not a possible possible as the join would not show from the representation of how it looks from the back angle originally shown. (i.e. the direction shown by the arrow)? The shaded areas are recesses. We have come to the answer B by elimination. This is a completely valid way of AB proceeding, but it would be useful to check that this is indeed the correct answer. The D recess shown in green on the right-hand side C of diagram B would be visible at the back of the lowest section of the three-dimensional E drawing. So B is not possible. Once again, this is more difficult than might be expected. We do not actually know what the hidden reverse side looks like – there are an infinite number of possibilities. All we can do is consider what hints are given by the three-dimensional picture. One primary feature of this type of question is that the answer cannot be produced just by considering the information given and the question. This is a backward question. The five options must all be looked at and a decision made on whether each is possible. Backward questions are a regular feature of questions on spatial reasoning, and of identifying similarity, which was dealt with in Chapter 3.7. They do not occur very often in the other types of question. The value of elimination was shown in the method of answering this type of question. 3.9 Spatial reasoning 113

Summary • This chapter introduced questions that are backward in that the answer must be found • We have seen the importance of spatial from the options rather than just from the reasoning in many occupations and how information and the question. problem-solving questions can test this. • The use of elimination in answering such • The value of practice in solving this type of questions was illustrated. question has been emphasised. End-of-chapter assignments 1 Fred wants to write the letters NSRFC  If somebody walks from X to Y, in how on his forehead for this afternoon’s many, and what, different orders will they Northampton Saints Rugby Football Club see the flagpoles? (Exclude places where match. He does it with face paints while one is exactly hidden behind another.) looking in a mirror. What should it look like in the mirror? 4 Our local café has an unusual clock which is upside down. The numbers 3 and 9 are 2 Draw a simple picture of your house or in their conventional places, but 6 and 12 another building with which you are familiar are interchanged. What time is it when the as seen from above and from the front. hands are positioned as in the clock face How much can you tell about the side and shown? back views from your drawings? 3 Outside the Diorama hotel there is a set of flagpoles, as shown in the drawing. The flagpoles are all painted different colours (red, blue, yellow, green, orange, white). RY O BG W XY A 2.45   B 7.15   C 8.45   D 10.15   E 10.45 When they are seen from position X, they are seen in the order (from left to right): R B Y O G W. 114 Unit 3 Problem solving: basic skills

5 The solid shown, which is a cube with two 6 Some children are making decorations. A corners cut off, is made from a shaped square sheet of paper is folded along a and folded piece of cardboard. (The dotted diagonal and then again so the two sharp lines represent edges which are hidden.) points meet, as shown. A cut is made through all the layers of paper along the dotted line shown and the small pieces removed. The paper is then opened. Which of the following pieces of cardboard What does it look like? (Try to visualise it in will fold to make the shape? There may be your head before you make a model to test any number correct from none to four. that your answer is right.) AB AB CD CD E Answers and comments are on page 321. 3.9 Spatial reasoning 115

3.10 Necessity and sufficiency Another type of problem involves identifying Activity whether there is enough data to solve the problem and, if not, which data is missing. I use the trip meter on my car to measure This is a useful building block in problem- the distance driven since I last had the car solving. It highlights one of the key elements serviced, so that I know when the next of problem solving, which is to find a way to service is due. The trip meter can be set to solve a problem without, in this case, having zero by the press of a button and records to do any arithmetic. the kilometres driven since it was last reset. The words ‘necessity’ and ‘sufficiency’ are I set the trip meter to zero after my used in mathematics but have exactly the last service. The next service is due after same meaning as they do in normal language. 20,000 km have been driven. Some time An individual piece of data is necessary to later, I lent the car to my brother. I forgot solve a problem if we cannot solve the to tell him about the trip meter; he problem without it. A set of data is sufficient pressed the button to zero it and drove to solve a problem if it contains all the 575 km. I then started driving again information we need. without realising what he had done. Identifying which data is needed to solve a What should the trip meter read problem can save effort in finding unnecessary when the next service is due? data or in making unnecessary calculations. Such questions are approached in a manner The above problem cannot be solved with the similar to those described in earlier chapters. information given. What additional piece of information is needed to solve it? To illustrate the type of question described here, we start with a very simple example. Commentary Suppose someone is taking a car journey. We This question is actually rather easier than it know their leaving time and we know the may at first seem. The distance driven from average speed they will do. We want to know the last service when my brother returned the their arrival time. Which other piece of car was the distance I had driven plus the information is necessary for us to calculate distance he had driven. I know how far he had this? driven, so what I need to know was the distance on the trip meter when I gave the car The solution is very straightforward: we to my brother. need the distance of the journey. We can then calculate the journey time (distance divided by In this case, like the previous example, we speed) and thus the arrival time. All of the were not asked to solve the problem, merely to three pieces of data we now have are necessary identify what pieces of information were to do this calculation. The three pieces taken needed to solve it. In real-life problem solving, together are sufficient. the data is not generally given; it has to be Here is a slightly more complex example. 116 Unit 3 Problem solving: basic skills

found. Having the skill to know which pieces Guineas Half-crowns Pennies of data are needed can save considerable time 2 3 10 and effort. Solving this type of problem does 2 4 9 not need particular mathematical skills – just 2 5 8 some clear and logical thinking. 2 6 7 3 4 8 Activity 3 5 7 4 5 6 I have a small collection of three types of old coin. The collection contains 15 coins in Of the options given, only C gives a unique total. There are more pennies than half- set. If there are 3 more pennies than half- crowns and more half-crowns than guineas. crowns, there must be 8 pennies, 5 half-crowns and 2 guineas. Which one of the following single pieces of information would enable you to know exactly Why do the other options not work? how many of each type of coin there was? Summary A There are 4 more half-crowns than guineas. • We have met a new type of problem where, rather than being asked to find a solution, B There are 5 more pennies than we are asked to find what pieces of guineas. information are necessary or sufficient to solve it. C There are 3 more pennies than half- crowns. • We have also encountered problems where we have to find a solution, but need to D There is one fewer penny than guineas identify an additional piece of information and half-crowns together. which is necessary either to help us with the method of solution or to choose Commentary between different possible solutions. In this problem, we are being asked to find which of the options is sufficient (along with • We have learned the meaning, in this the information we have already been given) context, of the words ‘necessary’ and to solve the problem. ‘sufficient’. There are 12 ways that 15 can be • We have seen various types of problem partitioned into three different numbers: which require extra data: some needing mathematical solutions; some only Guineas Half-crowns Pennies requiring us to establish a logical method of solution. 1 2 12 1 3 11 1 4 10 159 168 3.10 Necessity and sufficiency 117

End-of-chapter assignments assistant to count the bags. However, the assistant, not being very bright, counted 1 I have made a dice out of a sheet of the total number of pieces of fruit instead. cardboard in the form of an octahedron, George was about to send him back to which has eight faces as shown below. repeat it when he realised that the number that the assistant had given him was not I now want to number the faces from only sufficient information for him to work 1 to 8. The numbers on opposite faces out how many bags there were of each, but must add up to 9, so when I number a face was also the maximum such number. How 1, the opposite face must be 8. many bags of pears and bananas were there?  If I start with number 1 and work up, 3 Kuldip told me she had 12 coins in her how many faces can I number before I am pocket, all either 1¢, 2¢ or 5¢, with a left with no choice about where to put the different number of each denomination. numbers? There were more 2¢ than 1¢ coins and more 5¢ than 2¢ coins. She asked me 2 (Harder task) George stocks bags of how much money she had in her pocket pears and bananas in his shop. Each altogether. I told her that I did not have bag contains either five pears or three enough information to answer. bananas. He wanted to know how many to  Which of the following additional pieces order to keep his stocks up, so he sent his of information would enable me to know how much money she had in his pocket? A  She had three 2¢ coins. B The total amount of money was a multiple of 10¢. C 5¢ coins amounted to 34 of the total value. D She had two more 5¢ coins than 1¢ and 2¢ together. Answers and comments are on pages 321–2. 118 Unit 3 Problem solving: basic skills

3.11 Choosing and using models The most obvious and familiar use of the word consist of large numbers of equations and ‘model’ is that of a replica of an object, for associated data, and are implemented on example a car, at a smaller scale. In this book computers. They can predict (with varying the word is used in a wider sense. Models can success) things such as what will happen to be pictures, graphs, descriptions, equations, the inflation rate if interest rates are raised. word formulae or computer programs, which Such models are gross simplifications because are used to represent objects or processes. there are too many variables contributing to These are sometimes called ‘mathematical the condition of a national economy and all models’; they help us to understand how factors can never be included. things work and give simplified representations that can enable us to do ‘what Scientists also use models, for example in if?’ type calculations. predicting population growth. Such a model, for example, to predict fish stocks in fishing Architects, for example, use a wide range of areas, can be invaluable as it may be used to models. They may build a scale model of a control quotas on fish catches to ensure that building to let the client see it and to give a fishing does not reduce stocks to better impression of how it will look. Their unsustainable levels. drawings are also models of the structure of the building. In modern practice, these In both of these cases, the model has been drawings are made on a computer, which will produced as a result of a problem-solving contain a three-dimensional model of the exercise. The actual development of a model to building in digital form. This may be used to represent a process is beyond the multiple- estimate material costs and carry out choice questions in the lower-level thinking structural calculations as well as producing a skills examinations and will be dealt with in three-dimensional ‘walk-through’ picture on Chapter 5.2. Multiple-choice questions on the screen. choosing and using models test some of the basic skills involved in modelling and the This chapter deals with the recognition and extraction of data from mathematical models. use of appropriate models. A simple example of a model is a word formula used to calculate In the following activity you are asked to cost. The amount of a quarterly electricity bill use different models to compare calculations. can be described as ‘A standing charge of $35 This example is close to a real-life situation. plus 10¢ per unit of electricity used’. This may equally be shown algebraically as: Activity c = 35 + 0.1u The current structure of income tax collection in Bolandia is that the first $2000 of annual where c is the amount to pay (in dollars) and earnings are tax-free (this is called the tax u is the number of units used. threshold), then 20¢ of tax is charged on every dollar earned over this (this could also A more complicated example of a model be described as a 20% tax rate). would be the type that governments set up to simulate their economies. These usually 3.11 Choosing and using models 119

The government is determined to reduce mathematics. What effect would these the tax burden on lower-paid people and changes have on someone earning $50,000 intends to bring in a new system, which will a year? mean that the threshold for paying tax will rise to $10,000. They intend that all those As an exercise, consider other possible tax below the average earnings of $26,000 will structures which might give similar results. pay less tax, and all those earning more than Plotting graphs of tax paid against earnings this will pay more tax. What will be the tax gives a clearer representation of how the rate on earnings over $10,000? various models of taxation work. The graph below shows the tax paid under the current Commentary system of tax in the example above. The model of tax used here is quite simple, consisting of a fixed amount of income on which Tax due 12,000 no tax is paid and a single standard rate on 10,000 earnings above this amount. Currently, those 10,000 20,000 30,000 40,000 50,000 earning $26,000 pay (26,000−2000) × 0.2 dollars, 8000 Annual earnings or $4800. If they are to pay the same under the 6000 new regime, they will pay tax on $16,000, and 4000 the total will be the same as before, i.e. $4800. 2000 The new tax rate will be 30¢ on each dollar earned over $10,000 ($16,000 × 0.3 = $4800). 0 This could be done by algebra, but the process 0 is no simpler than that given above, which requires no more than elementary school Add lines for the proposed new system and for any other model you may think of. The next activity, below, requires you to go some way towards developing a mathematical model of a new situation in order to solve the question. Activity There are a number of different charging structures they could use. All taxis charge a My company regularly uses a taxi service to fixed price per kilometre and per minute of take staff to the airport. If there are several journey time. In addition they may charge a passengers needing to travel from our town fixed hire fee and an additional charge at similar times, they combine this into a depending on the number of passengers single journey. They divide the total cost by carried. the number of passengers and invoice each passenger separately. The distance is always What is the charging structure used by this the same and the time only varies by a small taxi company? What limitations are there to amount, but I do not know how they work out the conclusions we can derive? the charge for the journey. Number of passengers 1 2 3 4 5 Charge per journey per passenger $40.00 $19.98 $14.68 $12.03 $10.38 120 Unit 3 Problem solving: basic skills

Commentary A graph can be a very useful tool for If we just look at the data as it stands the analysing data such as in the table below, and pattern is not clear, other than that the price can also help in developing models. Try per passenger drops with the number of graphing the data, for both the cost per passengers. Since we are looking at the charge passenger and the total cost per journey. Does made by the taxi company, it is preferable to this help in clarifying the charging structure? look at the total cost of the taxi in each case. This may be carried out by multiplying the The activity above introduced the idea that cost per passenger by the number of models usually are approximations to the real passengers, as shown in the table below. world. The model used did not allow for variations in the time of the journey. This is The pattern now becomes much clearer. why the word ‘model’ is used. Almost all Allowing for some small variations (it was stated models are approximate – the model car does that there was a small variation in journey not usually have an internal combustion time), the first two values are the same and they engine. Economic models cannot take into then increase by $4 per passenger. We can, account factors such as the weather. therefore, conclude that the taxi company hire fee includes one or two passengers, then there is Many people use models in their everyday an extra charge of $4 per additional passenger. lives without even realising it. An efficient The $40 ‘basic’ fee covers the hire charge, the shopkeeper will, for example, have a set of distance charge and an average time charge. We rules that tells her how much ice cream to have no information which will enable us to order so she has plenty in the summer months separate these three items. A model can only be and less stock in the winter. as good as the data on which it is based. Number of passengers 1 2 3 4 5 Charge per journey per passenger $40.00 $19.98 $14.68 $12.03 $10.38 Total charge per journey $40.00 $39.96 $44.04 $48.12 $51.90 Summary • We have used real data to calculate the constants used in a mathematical model, • We have learned how a mathematical for example the starting rate and charge or graphical model may be used to per mile for a taxi fare. approximate real-life processes. • A graph of any sort is a model from which • We have seen how models can be used it is possible to get a picture of how to simulate changes in cost structure and variations can occur. their effects. 3.11 Choosing and using models 121

End-of-chapter assignments 2 Finn walks to school, a distance of 1.5 km which takes him 15 minutes. His older 1 A novelty marketing company is selling sister, Alice, cycles to school on the same an unusual liquid clock. It consists of two route at an average speed of 18 km/h. tubes as shown. The right-hand tube fills She leaves home 5 minutes later than up gradually so that it is full at the end Finn. Does she overtake him on the way of each complete hour, and then empties and, if so, where? At what time would she and starts again. The left-hand tube does have to leave to arrive at school at exactly exactly the same in 12 hours. The time the same time as Finn? shown on the clock is 9.15. 3 A shop normally sells breakfast cereal for  Draw what the clock looks like at 4.20. $1.20 a packet. It is currently running a promotion, so if you buy two packets, you get a third free.  Tabulate and graph the price per packet for numbers of packets bought from 1 to 10. How would other special offers (e.g. ‘Buy one, get one half price’) affect the shape of this graph? If you were working backwards from the graph, how could you determine which offer is currently being used? Answers and comments are on pages 322–3. 122 Unit 3 Problem solving: basic skills

3.12 Making choices and decisions Many of the problems we encounter in A  is 2 × $1.29 = $2.58 for 300 g everyday life involve making choices and B gives 400 g for $2.89, so 300 g is 0.75 × decisions. To buy or not to buy? Which one to buy? How much to buy? Which train to $2.89 = $2.17 take? All these are types of choice and C gives 600 g for 1.5 × $3.36 = $5.04 or decision that contribute to problem-solving processes and involve the use of skills that 300 g for $2.52 can be tested by problem-solving questions. D first 150 g is $1.57, second is $1.07, so These questions may involve skills that have 300 g is $2.64 been covered in earlier chapters: extracting E $1.57 for 200 g or $2.35 for 300 g information, processing data and finding methods of solution. The only real difference is So B is the best value. that the question asks for a decision to be made. This is quite straightforward; no skills that The following is an example of such a question. have not already been introduced are Activity involved. It is just necessary to work efficiently and correctly, finding the most effective way My local shops all have different discounts of approaching the problem. on jars of coffee. Which of the following represents the best value for money? The activity below involves making a decision. A  Everlo: $1.29 for a 150 g jar Activity B Foodland: $2.89 for 200 g, buy one Eve buys chickpeas in bulk to sell in her shop. get one free They come by volume and each drum can C Springway: $3.36 for 300 g, buy one contain between 10 and 12 kg. She repackages them to sell in 0.5 kg packs. She get the second half price gets a delivery on a Monday morning and sells D Superval: $1.57 for 150 g, 50¢ anything from 8 to 15 packs in a day. She is open 7 days a week and on Sunday night has voucher off the next 150 g (one 14 packs left and half a drum of bulk voucher per customer) chickpeas. E Massive: $1.57 for a 150 g jar with 50 g extra coffee free How many drums should she order to make certain that she has enough for the next Commentary week? In this case it is easiest to express all the prices to a 300 g equivalent – you may see A 2  B 3  C 4  D 5  E 6 that this requires fewer calculations than, for example, converting them all to 100 g. Commentary This is a maximum and minimum type problem: we have to combine the most 3.12 Making choices and decisions 123

chickpeas Eve could sell with the least she has Summary in stock and minimum in the drums she buys. • We have seen how problems may involve The least she has in stock is half a drum, making choices and decisions. which might be as little as 5kg; this will make up a minimum of 10 packs. She has 14 packs in • Decisions may involve selecting one item stock, so at least 24 in total. She needs at most from a number of options or making a 7 × 15 = 105 packs, so may need as much as 81 decision on an action. × 0.5 kg = 40.5 kg. At 10 kg per drum, she will need 5 drums to be sure she can last the week. • In solving these problems it is important to choose an efficient way of working You might also like to work out what is the so the correct answer is obtained and fewest drums she might need. calculations are carried out in the most effective and simplest way in order to This illustrates a particular type of decisions reduce the chance of error. question – where the decision is based on the minimum (or sometimes the maximum) to fulfil a criterion. End-of-chapter assignments 1 In a game of pontoon dice, you continue After four throws you have scored 17. throwing a single die until the sum of all Should you throw once more? Consider the your throws exceeds 21 (bust) or you chances of getting different scores and how decide to stop. You win plastic counters much you will win or lose. What is best on depending on the score you stop at. average? Stopping score Counters won or lost  What should you do if you have a score other than 17? 1 to 12 0 2 Clyde’s local supermarket has an offer 13 or 14 Win 1 on petrol, depending on the amount you spend in the store. If you spend $20–$30 15 or 16 Win 2 you get a voucher that gives you 2¢ per litre off petrol; if you spend $30–$50, you 17 or 18 Win 3 get 3¢ per litre off; and if you spend over $50 you get 4¢ per litre off. Clyde’s car will 19 Win 6 take 30 litres of petrol. 20 Win 8  Consider for what range of total purchase prices in the supermarket it is 21 Win 10 worth his buying a small amount extra, so that the reduction in the petrol cost will Over 21 Lose 4 make his total bill smaller. 3 Students at a school have to decide what subjects they are going to study next year. English, science and mathematics are all compulsory, but they can choose the remaining four subjects. 124 Unit 3 Problem solving: basic skills

The table shows how the choices can 4 The country of Danotia prints stamps in be made. Students must choose one the following denominations: subject from each column. The fourth subject may come from any column. 1¢, 2¢, 5¢, 9¢, 13¢, 22¢, 36¢, 50¢, $1.00, $5.00 12 3 A mail order company sends out equal geography French history numbers of three sizes of package, which incur postal charges of 34¢, 67¢ and technology German religious studies $1.43. They want to stock as few different denominations of stamp as possible (but art physical education not only 1¢ stamps as sticking lots of these on envelopes would be a nuisance!). music Latin  Which ones should they stock? Answers and comments are on page 323. Which of the following combinations would not be allowable? A French, geography, physical education, art B  French, German, Latin, music C  technology, German, art, history D  French, German, geography, music E geography, music, French, religious studies 3.12 Making choices and decisions 125

Unit 4 Applied critical thinking 4.1 Inference The verb ‘infer’ means to draw a conclusion, that I would be committing an offence if I usually from some factual (or supposedly tried to spend these banknotes. I would need factual) information. The information to know that passing false currency is illegal, provides the grounds for the inference. If the but that is such common knowledge that it grounds are good, we say that the inference is practically goes without saying. sound, or reliable. Another word that is often used is ‘safe’. If the grounds for an inference It would be fairly safe, too, to infer from [1] are poor, we have to say that the inference is that unsafe: it cannot be relied upon. [3] The banknotes are forgeries, Judging whether an inference is safe is therefore similar to judging whether an on the reasonable assumption that only argument is sound. The main difference is that forgeries could have duplicate numbers. By in a standard argument the conclusion is contrast it would be entirely unsafe to infer stated. In many texts and documents, that however, there is no explicit conclusion. There may be claims; there may be information. But [4] The banknotes are the work of terrorists, unless some further inference is drawn from intent on destabilising the economy. the information, there is no argument. Sometimes there may be an implicit Inferring [4] from [1] would be a blatant conclusion, where the information is clearly example of jumping to a conclusion. leading in one particular direction, and it is obvious what we are meant to infer. There is an Inference and science example in Chapter 2.4, item [7] (page 34). Here is another example (the subject may Drawing inferences, and judging the reliability sound familiar): of inferences, are especially important in scientific contexts. Scientists typically base [1] These banknotes all have the same their claims to knowledge on the information serial number. All genuine banknotes they collect from observation and experiment. have different numbers. Because we cannot confirm the truth of such claims without supporting evidence, we have This time the conclusion has been left unsaid. at least to be sure that the evidence is strong But it is clear what the author is getting at. If and the inference is reliable. Assessing what asked what can be inferred from [1] most can and cannot be inferred from a given people would probably answer that: document is therefore a key component of critical thinking. [2] The banknotes are not all genuine. Here is a short introductory example: Moreover this would be a safe inference, because [1] provides very good grounds for [2]. DOC 1 Of course [2] is not the only inference that Ice ages last for roughly 100,000 years, could be drawn from [1]. Nor is it the only safe going by the record of the past half-million one. [1] also gives good grounds for inferring years. The warm phases in between are called interglacials. The standard view, until quite recently, has been that we are 126 Unit 4 Applied critical thinking

coming to the end of the present warm Commentary phase, which has already lasted just over We’ll consider the inferences in turn. The first 10,000 years. Indeed, data from Antarctic one, A, as well as being a bit vague, has little ice cores* indicated that the previous support from the text. If we take ‘any three interglacials have lasted between millennium now’ to mean in the next one or 6000 and 9000 years which, if repeated, two – which is its natural meaning – then A would have seen parts of Europe, Asia and would mean sticking with the standard view, North America covered in ice since before despite the most recent findings casting doubt the rise of the Roman Empire. The most upon it. The standard view is that the present recent Antarctic ice cores have revealed warm phase should be reaching its time limit; that the warm phase before that lasted for but according to the last three warm phases the 30,000 years. It is known, too, that the limit has already been passed. The latest ice core Earth’s alignment relative to the Sun during also suggests that some interglacials could last that long interglacial was similar to its as long as 30,000 years. Evidence that the alignment at the present time. Earth’s present alignment with the Sun’s rays resembles that of the last long interglacial, * An ice core is a sample obtained by drilling pours even more cold water on A. (Interestingly, down into the ice cap. The state of the ice at even without the most recent evidence, the different levels provides a climatic record that grounds for A would still be weak: half-a- can extend over hundreds of thousands of million years is a blink of an eye in geological years. terms, and the sample of just three warm phases is too small to call a reliable trend.) Activity B might seem consistent with what has just From the information in Doc 1, which of the been said about A. If we are not near the start of following can reliably be concluded? a new ice age, the standard view must be wrong. But there is a lot of difference between A Another ice age is due any millennium saying that an inference is unsafe, and now. declaring it false. B is a much stronger claim than can be supported by the data in Doc 1. We B The standard view is wrong. might be near the end of a warm phase: one C The present warm phase is set to last that is longer than the last three and shorter than the one before that. There is little or no another 20,000 years. positive evidence for such a claim; but nor is D According to the recent geological there proof that it is false. Remember the significance of strong and weak claims (see record, ice-age conditions are the norm, Chapter 2.2). A strong claim requires much and it is Earth’s present climate which more to justify it than a more moderate claim. is unusual. Had B asserted that the standard view is now E Global warming is delaying the start of less plausible than it was, instead of plainly the next ice age. false, that would have been defensible. Give a brief reason for each response. C suffers from the same fault as B: it, also, is too strong. A single example of a 30,000-year 4.1 Inference 127

warm phase, coinciding with a particular Why do we use the word ‘safe’? alignment of the Earth and Sun, is far from adequate to justify an out-and-out prediction The practice of calling some inferences unsafe that the present interglacial will also last that is a recognition of the importance of long. No serious scientist would go so far. The reasoning carefully. What makes an inference author of the Geological Society webpage unsafe is not just that it may be wrong, but from which the information is Doc 1 was that it may have consequences, sometimes sourced claims only that: very serious ones. Perhaps the most obvious illustration is a criminal trial, where a verdict F On these grounds, even without human must be reached on the basis of the evidence. intervention, another 20,000 years of A trial-verdict is a particularly serious kind of warmth may be expected. inference, on occasions a matter of life or death. As the result of a faulty inference, an Compare this with C. Although it is drawing innocent person might go to prison for a long broadly the same conclusion, it expresses it in a time. On the other hand, a not-guilty verdict careful way which would permit its author to passed on a guilty person may leave him or her fend off objections. Another 20,000 years may be free to commit a further atrocity. You will expected, on the evidence supplied. That claim sometimes hear the expression, ‘That’s a does not cease to be true, at the time of making dangerous inference to make!’ We can easily it, even if in 10,000 years’ time it turns out to see why that is entirely appropriate. have been optimistic. Nor will F be falsified if new, contrary evidence comes to light, because But even if there are no obvious dire the author has qualified the claim by prefacing it consequences, it is still important to reason with: ‘On these grounds . . .’ You should well rather than badly, because it gets us closer remember these details of presentation when to the truth. Judging when an inference is safe you are constructing your own arguments. They or reliable is therefore a key element in critical may seem like purely linguistic points; but they thinking. can make the difference between inferring something that can be substantiated, and Assessing inferences something that cannot. With this in mind, read the following passage Whoever inferred D has taken the right sort of (Doc 2) and make a mental note of the care in expressing it. It is preceded by the phrase: information that it contains. It introduces a ‘According to the recent geological record . . .’ topic which will occupy the rest of the and draws a conclusion that the record firmly chapter, and feature in the next two. supports. Recent ice ages (geologically speaking) have lasted between 3 and 17 times longer than www.CartoonStock.com interglacials. So glacial conditions have been the prevailing ones during that period, and it is the Earth’s present climatic state that is unusual. D is a safe inference. For E to be true it would mean that the standard view was correct after all, and/or that the trend of the last three glaciation cycles was continuing. It would also require global warming to be taking place; and to be capable of delaying the onset of an ice age. E is not impossible, but it would take a lot more than the claims in Doc 1 to make it true. In a word, it is unsafe. 128 Unit 4 Applied critical thinking

DOC 2 A Lawyers are self-serving and unscrupulous. The industry surrounding large compensation claims following accidents B Without the no-win no-fee option there and personal injury has attracted much would be fewer claims for personal media attention, with stories of millions of injury. dollars being awarded in damages to successful claimants, and of C Advertising by law firms and/or claim compensation being sought for every kind management firms encourages clients of injury or loss. There has also been a to exaggerate or invent injuries. dramatic increase in law firms – or ‘claim management firms’ that act as D A no-win no-fee arrangement benefits intermediaries – canvassing for accident the lawyer much more than it does the victims by advertising on television, the claimant. internet or on the street. (The phrase ‘ambulance chasers’ is often used to E As long as they win more than 50% of describe them.) People complain of getting conditional-fee cases, a law firm need unsolicited calls or text messages asking: not be out of pocket. ‘Have you had an accident that wasn’t your fault? . . .’ or words to that effect. F Claims being pursued for personal Television – especially daytime television – injury have increased significantly has a high proportion of such ads. since the introduction of conditional- fee agreements. At the same time it has become commonplace for lawyers to offer their Commentary services on a no-win no-fee basis, which Although the passage has a somewhat allows people on low or moderate incomes negative tone it is not openly judgemental in to go to court with no risk of running up what it claims. We have to be careful, expenses they couldn’t otherwise afford. therefore, what we read into the passage and This is also known as a ‘conditional-fee what we infer from it. If someone has a poor agreement’. Lawyers earn nothing for opinion of lawyers generally, or of ‘ambulance unsuccessful claims but are entitled to chasers’ in particular, it would be easy to be led charge up to twice their normal costs if the by prejudice into thinking that this document claim is successful. This practice, known supports those viewpoints. It does not. as ‘uplift’, can add thousands to the bill the losing side then has to pay out. You cannot, for example, infer from it that lawyers are bad people, even if you happen to Activity think that they are. If you included A in your list of safe inferences, then either you weren’t For each of the following statements in turn, taking the question seriously, or you were assess whether or not it can reliably be simply giving a view based on prejudice, or concluded (inferred) from the above other data that is not provided here. In fact passage. there is practically never sufficient hard evidence for any claim as strong, as sweeping, or as judgemental as A. A is not even the kind of claim that can be ordinarily inferred reliably from purely factual information. What about B? This certainly seems a more reasonable inference, and a less opinionated one. There is, too, a widely held belief that the 4.1 Inference 129

number of claims has risen significantly since Note that even if there were statistics showing no-win no-fee agreements were introduced, that since lawyers started advertising, more and probably because of them. Since dishonest claims have been lodged, that would conditional-fee agreements give people on low not permit an inference that the advertising was or middle incomes the chance to pursue the cause, or that it gave encouragement (see expensive legal actions at no financial risk to cause–correlation fallacy in Chapter 2.10). In themselves, it is natural to think that there Doc 2, however, there is not even a correlation. would be a surge in claims. But the very fact We are not given any data on numbers of that B seems so reasonable, and happens to be dishonest claims before or after advertising widely believed, is precisely why we need to began. C is definitely not a safe conclusion. approach it critically. If you already assume that no-win no-fee arrangements have resulted D claims that no-win no-fee arrangements, in more personal injury claims, you are likely which on the surface look quite advantageous to see the passage as grounds for believing it. for the client, are actually of more benefit to the But interpreted neutrally, the passage neither lawyers. Assuming that ‘benefit’ means supports B nor disputes it. All that we are told financial benefit, it is clear from Doc 2 that in Doc 2 is that there are no-win no-fee lawyers do get some benefit from the way in arrangements on offer, and how they work. We which the system works. They may lose money are also told that this has prompted stories in when the case is unsuccessful, but they have the media, but with no comment on the truth bigger costs awarded when they do succeed. or falsity of these, or even what they actually Provided they win more cases than they lose, claim. There is no information about the effect they should be better off than if they didn’t the arrangements have had on numbers or take the case at all because the client could not attitudes. If we stick faithfully to what is afford the fee. On the other hand, the client contained in Doc 2, B must be seen as leaping benefits too, either by winning, or by having to an unjustified conclusion. nothing to pay if the case fails. The question is whether the lawyer benefits more than the C, likewise, may seem like a very believable client; and again we find the passage consequence of advertising for victims, uninformative. There is simply no data in Doc 2 especially if the advertisements emphasise the by which to quantify the gains comparatively. possibility of making big money out of an accident. Obviously, the worse the harm that However, Doc 2 does give strong support to has come to the claimant, the more money E. This is because the information in Doc 2 is the court is likely to award in damages. So mainly explanatory. In particular it explains there would be a temptation for a dishonest how lawyers can afford to take on cases without person to cheat by exaggerating or inventing charging a fee. When they win a case they get an injury. But that is very different from back around twice their normal costs, to make saying, as C does, that the advertising by law up for the fee they would have been paid by the firms and intermediaries encourages cheating. client, win or lose, under the old system. It is That is a serious allegation. It is also a little simple mathematical fact that so long as they hard to believe, if it is taken to mean that don’t lose more cases than they win, they are lawyers actively encourage dishonesty in the not out of pocket. If they win more cases than way they advertise. But even if we interpret C they lose, they make money. We cannot infer more charitably to mean that the advertising that they do better out of this arrangement has the unintended effect of giving some people than the clients, as claimed by D. But if Doc 2 is the idea of cheating, there is still no evidence factually correct, we can quite safely infer E. of any such connection in the passage. If you were really alert you might have added that E carries the implicit assumption 130 Unit 4 Applied critical thinking

that a lawyer’s cases all have the same value, quite trivial car accidents. The writer Andrew and require the same level of input in terms of Malleson wrote a book called Whiplash and hours worked, and other costs. Arguably a Other Useful Illnesses. The content is serious, lawyer could lose a small number of very but the book’s tongue-in-cheek title tells its complex cases and still be out of pocket own story. because they were worth more in fees than the income generated from cases he or she won. Here is another document, this time Strictly speaking E is safe to infer only if on graphical. It consists of three bar charts. (See average all cases cost about the same to pursue. Chapters 3.7 and 5.4 for further questions of this type.) The data is from an official questionnaire And so we come to F: ‘Claims being pursued conducted among 509 randomly selected adults for personal injury have increased significantly living in the UK. However, it probably reflects since the introduction of conditional-fee opinion in many developed countries. agreements.’ By now it should be clear that this cannot be inferred from Doc 2 either. You may DOC 3 have thought that F could be inferred because it seems so likely to be true, given that conditional- CHART 1 fee agreements allow people to make claims without having to pay anything. In that respect Compared with ve years ago, do you think there F is like B. But F can only be inferred if it is also has been a change in the number of people receiving assumed that there have been no other changes which might have had a reverse effect. Nor can F compensation payments for personal injuries? be inferred without assuming that lawyers now % take on more cases than they did before the introduction of no-win no-fee. Doc 2 provides A lot more people 67 no information to justify either of these receiving payments assumptions. In fact, the note about E in the previous paragraph suggests that lawyers may be A few more people 20 much more selective than they were, since they have more to lose. F could, quite realistically, be Virtually no change 5 false, and public opinion seriously flawed. Fewer people 1 The clear warning to take from the * Respondents who answered ‘Don’t know’ have been discussion above is that many seemingly excluded from chart. reasonable inferences were in fact unsafe. Some of the claims may be true, and may be found to CHART 2 be so after further investigation. But Doc 2, as it stands, lacks the hard data we would need for Compared with ve years ago, do you think there has drawing conclusions such as B, C or F. been a change in the number of people making false claims? Popular opinion % Many people hold the opinion that there is a A lot more people 50 growing ‘compensation culture’, with many making false claims more claims being made for injuries – real or otherwise – than there were, say, five or ten A few more people 29 years ago. Many also take the view that a lot of the claims are bogus, or fraudulent, especially Virtually no change 8 the infamous ‘whiplash’ injury, following Fewer people 2 * Respondents who answered ‘Don’t know’ have been excluded from chart. 4.1 Inference 131

CHART 3 ‘I might be tempted to make an exaggerated claim for a personal injury, even if I didn't have a strong case for compensation.’ To what extent do you agree or disagree with this statement? Disagree strongly Disagree slightly Neither Agree Agree % %% slightly strongly % % All adults 64 12 8 8 5 Those who watch daytime TV 67 13 7 7 5 Those who have recently 65 12 8 8 9 5 seen a personal injury claims ad *Respondents who answered ‘Don’t know’ have been excluded from chart. Take some time to think about and/or discuss Commentary the following questions, before reading the The first point to make is that the data commentaries that follow. concerns public opinion. The first of the three claims is therefore clearly supported by the Activity data. (As a matter of interest, it is the conclusion that the researchers drew 1 Which of the following are supported by themselves.) Only 1 in 20 people thought that the information in Chart 1? (The answer there had been no change. Well over half may be any, none or all.) thought that the change was substantial in favour of more people receiving payments. Assume that the data is accurate and that This indicates a ‘widespread and strong belief’, the sample of people questioned is and makes A a safe conclusion. The question representative of the population. of whether the sample was representative need not concern you, as you were told to assume A There is a widespread and strong that the figures are accurate and well belief that more people are receiving researched. compensation for personal injuries now than five years ago. Claim B is a more direct interpretation of the data, and simple arithmetic shows that it B 87% of those responding to the too is a safe inference. Claim A is true because survey believe that there are more claim B is true. people receiving payments for personal injury compensation than Claim C is more complex and more there were five years ago. interesting. It would seem reasonable to argue that if there are more people receiving C Claims being pursued for personal payments, there are more claims being made. injury have increased significantly in But ultimately C rests on the assumption that the past five years. 132 Unit 4 Applied critical thinking

there really are more payments being made; putting in false claims, and that would or, in other words, that the widespread belief explain the rise in the number of claims expressed by those questioned is correct. It is generally. this step which is the problem: reasoning from the evidence that most people believe These are plausible hypotheses, but they are something, to the conclusion that it is true or poorly supported by the data in the charts. The probable, is a classic fallacy known by the fact that one thing would explain another if it Latin argumentum ad populum. If you prefer were true, does not not permit us to infer that more modern names there are plenty to it is true. In some circumstances this can be a choose from: appeal to popular opinion, powerful argument; but as you discovered in appeal to consensus, appeal to the majority, Chapter 2.10, it can also be a dangerous one. the authority of the many over the few. The (Remember the Bayside fish restaurant. Just weakness of this argument is nicely captured because food poisoning would explain why a by the old joke that 40,000 lemmings can’t all restaurant has closed, it does not follow that be wrong. (The joke is that every so often food poisoning occurred or came from the whole colonies of lemmings are believed to restaurant.) In the present case, just because run to the edge of the nearest cliff and plunge cheating would explain rising claims, it does to their deaths!) C therefore is not a safe not mean there is widespread cheating. There inference. are many other equally plausible reasons why claims could be increasing in frequency, if Activity they are. 2 Suppose the majority view represented Activity in Chart 1 is correct. Would it follow that there has been a change in the number of 3 Does the data in Chart 3 contradict the people making false claims? data in Chart 2? Commentary Commentary This is another complex question. What A comparison of Charts 2 and 3 is very makes it so is that it is hypothetical. We don’t interesting. According to Chart 2, most people know whether the opinions represented in evidently believe that there is an increased level Chart 1 are true or not. The question is: If the of dishonest claiming going on, and half of sample of public opinion is right and there has those questioned believe that there is a big been a big increase in claims, can we infer that increase. But if Chart 3 is anything to go by, very a significant number of people are making few people say that they would so much as false claims? exaggerate a claim. Even those who watch daytime TV (which, we were told, carries a lot of Why might this be true? Well, if it is advertising by the so-called ‘ambulance correct, as it is widely believed, that there are chasers’), or who have seen a claims more people getting money for injuries, advertisement recently, say they are no more others may see this as a way of getting some likely to claim than the general sample of the money themselves. It is a sad fact that there population. Remember, too, that being tempted are dishonest people who will seize such to do something and actually doing it are two opportunities. Then again, it might be the different things. Of the 13% of all adults who other way round: that with the help of no-win said they might be tempted at all, how many no-fee agreements more people have begun would have gone as far as making a false claim? 4.1 Inference 133

We simply cannot say. We also have to wonder claims. But even if that widespread belief whether those questioned in the Chart 3 (shown by Chart 2) is unfounded, it is not survey would all have answered truthfully. because the data in Chart 3 contradicts it. It is perfectly rational for people to hold the There is no contradiction here. If 13% of all two views: (1) that there are more cheats than adults are willing to admit to being tempted to there were; but (2) that they wouldn’t cheat exaggerate a claim, that could result in a themselves. These do not conflict; so nor do considerable and increasing number of actual the two sets of data. claims. It could certainly be a sufficient number to explain why many believe that Before answering the next question, there there was a significant increase in dishonest is a further short document to read: DOC 4 How did no-win no-fee change things? Ten years after the introduction of no-win no-fee agreements the UK Compensation Recovery Unit reported that the number of cases registered to the unit had remained relatively stable. In 2000/1 there were 735,931. The number in 2007/8 was 732,750. For example, clinical negligence cases notified to the unit fell from 10,890 in 2000/1 to 8872 in 2007/8. Accidents at work cases fell from 97,675 in 2000/1 to 68,497 in 2007/8. Only motor accident claims have risen rapidly, rocketing from 403,892 cases in 2004/5 to 551,899 cases in 2007/8. In its 2006 report on the ‘compensation culture’,the House of Commons Constitutional Affairs Committee heard evidence that personal injury claims had gone up from about 250,000 in the early 1970s to the current level, but that the introduction of no-win no-fee had coincided with this levelling off. Lawyers dispute the claim that no-win no-fee inevitably leads to more frivolous claims and more cases generally. They say the solicitor acts as a filter, knowing that every case that doesn’t make it to court or a settlement is a financial loss to the firm. BBC News Magazine 134 Unit 4 Applied critical thinking

Activity However, Doc 4 also reveals that the small reduction in claims generally contrasts with a 4 Can it reliably be concluded from the massive increase in motor accident claims in information in the three charts and Doc 4 particular. If that rise is the result of an that public perceptions about false or increase in false and exaggerated claims, then exaggerated compensation claims are the public perception could be justified. A seriously mistaken? good answer to the question will therefore recognise that the evidence is inconclusive, Commentary with some of the facts pointing in one This is a more open question than the others, direction, some in another. This need not stop and consequently there is more than one you drawing a conclusion, but it should not be direction that your discussion could have too strong or overstated. If you inferred that taken, and more than one decision you could the majority were clearly correct in their have reached. What matters most is not which opinions, or completely wrong, without answer you gave, but why you gave it; how you acknowledging the room for doubts, your interpreted the evidence. You could, for conclusion would be unsafe. example, have noted that there is something of a contradiction between what the majority Summary think (Chart 1) and the official figures (Doc 4, paragraphs 1 and 2). Those figures reveal that • When presented with a source of the number of claims overall has ‘remained information, whether in text form or relatively stable’, or even fallen slightly over numerical or graphical, we often draw the period in question, with examples of conclusions / make inferences. medical claims and work-accident claims both being down. If the total number of claims has • A ‘safe’ (reliable, sound) inference is one fallen, it seems groundless to infer that the that has strong support from some or all number of false claims has risen. You might of the available data, and is not obviously also have added the point, already made in the contradicted by other data. comments after Activity 3, that Chart 3 casts some doubt on the belief that false claims are • To be ‘safe’ an inference or conclusion soaring. Your answer could therefore have must be more than just plausible or been that the public perception is simply false. reasonable. It must follow from the data. 4.1 Inference 135

End-of-chapter assignments photograph was conclusive evidence that a crime was being committed. On the other 1 Based on the discussions you have had hand there is enough detail in the picture and the commentaries you have read, write to raise suspicions. Assuming that this a short answer to each of the following was a genuine action-shot, and not posed, questions. (These are good preparation consider these four possible accounts of for some of the questions in Cambridge what is happening in the photograph: Thinking Skills Papers 2 and 4.) a Does advertising, especially on daytime A The person on the left of the picture television, encourage people to make is snatching a bag from the shoulder dishonest claims for personal injury? of the other, and is about to run off b Does the statistical information in Doc with it. 4 on page 134 contradict the view that claims for personal injury are on the B The person on the left is attempting increase? to take something out of the bag. 2 How much can be inferred, reliably, from a C The person on the left has photograph such as the one here? accidentally made contact with the person on the right. Without some background information it would be unsafe to say that this D The two people in the picture are friends walking together by a lake or river. Then either decide which of the above explanations can most safely be inferred, or if you think none of the above is a safe inference, suggest one that is.  Write a short justification for your conclusion, based on clues you can find in the picture. (For convenience call the person on the left of the picture ‘L’, and the one on the right ‘R’.) 3 ‘It is believed in many countries around the world, including the UK, that there is a damaging “compensation culture”.’  How much support is there for this belief in Docs 2–4? Your response should take the form of a short written essay. Answers and comments are on page 323. 136 Unit 4 Applied critical thinking

4.2 Explanation In this chapter we return to an important before hearing the argument. That is what we concept that was introduced in Chapter 2.8, call a ‘conclusion’. Explanations work in the namely explanation. Explanation, like opposite direction: they take something that argument, involves giving reasons. But we know or just assume to be true, and help us explanatory reasons do not lead to to understand it. Explanation plays a very conclusions, as reasons do in arguments. important role in science; and it is easy to see why. One of the main goals of science – if not Examine the following short passages. the main goal – is to discover how and why things are as they are: what causes them, what [1a] Seawater is salty. This is because the makes them happen. Once we can fully river water that drains into the oceans explain something, such as the saltiness of flows over rocks and soil. Some of the seawater, we can go on to predict or infer all minerals in the rocks, including salt, sorts of other related facts or phenomena. dissolve in the water and are carried down to the sea. Need for explanations [1b] The river water that drains into the Explanations are particularly useful when oceans flows over rocks and soil. Some there is something surprising or puzzling that of the minerals in the rocks, including needs to be ‘explained away’; or where there is salt, dissolve in the water and are a discrepancy between two facts or carried down to the sea. Consequently observations; or where there is an anomaly in seawater is salty. a set of facts. (An anomaly is an exception: something unexpected or out of the ordinary.) These are both explanations. To be more If a patient’s blood pressure is being precise they are the same explanation, with monitored, and on a particular day it is much slightly different wording. Typically, higher or lower than on all the other days, that explanations tell us why something is as it is, would be classed as an anomalous reading, and or how it has come about. The explanation might well lead the doctor to look for or here consists of two reasons: (a) that rivers suggest an explanation. flow over rocks and soil; and (b) that the rocks and soil contain minerals that dissolve in the Here is an observation that would seem to water. These two reasons, between them, be at odds with [1a] and [1b]: explain a fact, the saltiness of seawater. But the saltiness of seawater is not a conclusion or [X]  River water does not taste salty. inference drawn from [1a] and [1b]. Most of us don’t need any argument to convince or We are told by the scientists that seawater gets persuade us that seawater is salty. We have the its saltiness from the rivers that flow into it. So evidence of our senses. We can taste it, which why can we not taste the salt in the river? is a good enough reason to take it as fact. Unless you know the explanation, there appears to be a discrepancy here: if one tastes This is the key difference between an so strongly of salt, why does the other taste argument and an explanation. Arguments are fresh? By analogy, if you poured some water meant to give us reasons to believe something from a jug (the river) into an empty bowl which we did not know, or were less sure of, 4.2 Explanation 137

(the ocean), and then found that the water in Suggesting explanations: plausibility the bowl tasted salty, but the remaining water in the jug did not, you would be right to feel In the previous example the explanation is puzzled. You would probably infer that there grounded on good scientific evidence. If there had been some trick, since fresh water cannot were any doubt about it, scientists could turn into salt water just by being poured! measure the minerals that are dissolved in rivers; they could test rainwater and confirm that it is Activity pure, and so on. But not all facts or happenings can be explained with the same confidence, Give a concise explanation for the fact that either because they are more complex, or rivers taste fresh and the sea salty. You may because there is limited available data. know the reason, in which case just write it down as if you were explaining for someone Science is not the only field in which who did not know. If you don’t know the explanation is needed to account for facts. reason, try coming up with a hypothesis; then Historians, for example, do not just list the do some research, on the internet or in the things that have happened in the past, any library, to find out if you were right. more than scientists just list observations and phenomena. Like scientists, historians try to Commentary work out why events happened, what their The scientific explanation is as follows. The causes were. For example, take the following water that flows into the oceans does not all piece of factual information: remain there. The sun’s energy causes it to evaporate, after which it condenses again and DOC A falls as rain or snow. The rainwater finds its way back into the rivers and carries more salt In October 333 BC, Alexander’s Macedonian down to the sea. This process goes on in a force confronted the Persian king Darius III continuous cycle (part of what is called the and his army at Issus. The Macedonians, ‘water cycle’). The key to the explanation is though more disciplined than the Persians, that when the seawater evaporates, it leaves were hugely outnumbered. Yet, surprisingly, in the salt and other minerals behind, so that the furious encounter that followed, it was over an extended period of time (millions of Darius’s massive force that fled in defeat, years) the salt becomes increasingly leaving Alexander victorious. concentrated in the oceans. The relatively small amounts of salt that dissolve in a volume This is neither an argument nor an of river water as it flows to the sea aren’t explanation. It is simply a series of informative enough to give it a taste. Besides, rivers are claims; a statement of historical fact. However, constantly being refreshed by new rain and it is a fact in need of an explanation because, as melting ice or snow. the text says, it is a surprising fact. Normally, if one side in a battle hugely outnumbers the The analogy of the jug and the bowl is other, the larger army wins, unless there is therefore a bad one. It misses out the key some other reason for the outcome. If the larger factors of evaporation and the large timescale. army wins no one is very surprised. No one is To explain why seas are salty and rivers are likely to ask: How did such a big army beat such fresh, you have to include the fact that the a small one? Usually it is only when the result is process has taken a very long time. unexpected that we want to know why. With this case, as with many other historical events, we don’t know for certain why or how Alexander turned the tables on Darius. But there are many possible 138 Unit 4 Applied critical thinking

explanations which, if true, would explain the Don’t jump to conclusions outcome of the battle, against all the odds. Alexander may have used better tactics. He You saw both in the previous chapter and in may have had better weapons. He may have Chapter 2.10 that some of the worst reasoning been a more inspiring leader than Darius. errors come from jumping to conclusions. The small numbers may have made the This is a particularly strong temptation when Macedonian army more mobile, easier to inferring causal explanations. Suggesting command. The Persians may have been tired, explanations is fine. Assessing their or sick, or suffering from low morale. They plausibility is fine. But just because an may have been overconfident because they explanation is plausible it doesn’t follow that had more soldiers and were taken by surprise it is true. If it were a fact that Darius had a huge by the ferocity of their enemy, and so on. One but poorly trained army, that could explain the or more of these possibilities could have been Persian defeat at Issus. But so might good sufficient to change the course of the battle training explain the Macedonian victory. away from the foregone conclusion that most people would have predicted. We cannot say Moreover, it might be neither of these. It which, if any, really was a factor, still less might be something quite unlikely. the decisive factor, on the day. All we can say Conceivably the battle was determined by with certainty is that there are competing Alexander’s mother casting a magic spell, or hypotheses. But we can make some valid laying a curse! This may seem fanciful and judgements: we can assess the competing implausible. We don’t really believe these days explanations in terms of their plausibility. We in spells and curses as real causes. But can ask, of a proposed explanation: Would it, sometimes the wildest theories turn out to be if true, have explained why the battle went correct. It is fairly well documented that in Alexander’s way? If the answer is yes, it is a ancient times people were much more plausible explanation, even though we cannot superstitious than they are today. Oracles and infer that it is the explanation. soothsayers were taken seriously and consulted before decisions were made; witches were Conversely, we can say that certain burned for the evil powers they were thought statements would not adequately explain the to possess. Had a spell been cast, and believed, outcome even if known to be true. The fact the psychological effect could have been quite that Alexander’s soldiers were Macedonian is potent. It might have filled one side with not an adequate reason, though it is a fact. It confidence, and/or the other side with terror. might be adequate if we also knew that Macedonians were particularly skilled or Judging alternative explanations ferocious or dedicated fighters; but on its own the fact of being Macedonian does not explain It is all very well to say that if something were their victory. Similarly, if we were told that true it would explain a fact. The mistake is to Alexander later became known as ‘Alexander move too quickly from the discovery of a the Great’, that would not explain the victory. satisfying and credible explanation to the It is his victories which explain why he was inference that the explanation is true. called ‘the Great’. Nor would the fact that Explanations need to be evaluated just as Darius’s soldiers fled when they realised they critically and carefully as the reasoning in an were beaten count as an explanation: it would argument. In the case of explanations we are just be another way of saying that they were looking for the best. What makes one better defeated, not a reason why. than another? There are two useful tests for judging the effectiveness of an explanation. One is to question its scope; the other its simplicity. The 4.2 Explanation 139

‘scope’ of an explanation is just shorthand for Category 2000/1 2004/5 2007/8 Change how much it can explain. Staying with ancient of claim history for a while longer, some serious defect among Darius’s troops, on that fateful October Clinical 10,890 8872 down day, could explain wholly why Alexander 2018 won, without requiring any extraordinary brilliance from his enemy. Perhaps half the Work 97,675 68,497 down Persian soldiers had dysentery; or there was a related 29,178 mutiny. These are singular explanations which, if true, would explain a singular event. Motor 403,892 551,899 up But if Persian weakness was the whole 148,007 explanation, it would be difficult to explain how Alexander’s elite force won so many other ALL 735,931 732,750 down battles, across most of the then known world, CLAIMS 3181 and against armies that frequently outnumbered them. By most accounts he was What would explain this? Why would one never defeated (at least until he reached India, category of claim have risen sharply (up by the limit of his empire). It is highly implausible 37% in just four years) when all the others that each time there was some different, declined? One obvious answer is that the unique reason for victory. number of accidents had risen. If that was true it would certainly be a plausible explanation, Far more plausible is that Alexander, and/or and therefore a reasonable hypothesis. his army, was immensely talented. We say that this explanation has ‘scope’, because as well as The trouble is, it is not true. An official explaining the outcome at Issus, it explains government report in 2011 states that: countless other victories. Nor does it require his enemies to be defective: if Alexander was DOC B superior that was enough. Someone who . . . over the past two decades, the number wished to detract from his achievements might and severity of accidents has reduced. come up with a different explanation for each Compared with the 1994–98 average, in 2010 of his successes, always suggesting there was there were fewer people killed or seriously some failure in his opponents. But the standard injured in road accidents (−49%) . . . and, the historical claim, that he was an amazing slight casualty rate was lower (−39%). general and brilliant tactician, is a far simpler account, as well as explaining much more. Plainly the hypothesis is dead in the water. If the number of accidents explained the Anomalies number of claims, the trend should have been down as sharply as it was in other categories! Look again at Doc 4 in Chapter 4.1 (page 134) about compensation claims for injuries. In the Activity second paragraph of the document we read that out of all the different kinds of claim only Suggest and assess one or more alternative motor accident claims have risen; all other explanations for the anomaly shown in the categories fell. That is to say, motor accident table above. claims represent an anomaly: they ‘buck the trend’. If a table were created to match the data Commentary for the years in question, it would look like the There are various explanatory avenues following. which can be explored. One is that people really are faking or exaggerating injuries, and in very large numbers. Another is that although there were more accidents in the 140 Unit 4 Applied critical thinking

past, people were not bothering to claim. Yet can turn out to be factually untrue, thus ruling another is that advertising by law firms and it out. others has encouraged people to claim who would not have done so in the past, because Here is another interesting example. The they would not have thought they had a English word ‘posh’ is widely believed to be strong enough case. The trouble with all of an acronym, P.O.S.H, formed from the phrase: these is that we still have to explain why claims are down in all but this one category Port Out, Starboard Home of motor accidents. So below are two suggestions. (There may be other plausible This phrase, it is claimed, dates back to the suggestions, besides these.) 19th century, when people travelling to India and the Far East would normally go by sea. Suggestion 1: To make a claim for an injury, Wealthy European passengers, it was said, you have to be able to pin the blame on demanded the more expensive cabins on the someone else. In the case of a motor accident, port side of the ship travelling east (out), and it is usually quite easy to prove whose fault it on the starboard when returning (home), is. (It may be very much harder to prove that a because they were cooler in the hottest part of doctor or employer was negligent.) So the day. The request was allegedly written on claimants, or their lawyers, go for the easiest the tickets of these passengers using the initials type of claim. That’s a possibility. only. Hence the word ‘posh’ entered the language as a description for persons of wealth Suggestion 2: The public are very aware of and position who could afford such a luxury. the incidence of road accidents. Most people have either seen one, experienced one, or It is a very satisfying, pleasing theory, and know someone who has had one. Because the one which seems too plausible to be wrong. other categories of accident are less common, However, there is not a shred of hard evidence there is less awareness of them, and so people for it: tickets, for example, with the initial letters are less likely to think of making a claim for, on them. Most experts (lexicographers, say, a workplace accident or poor medical etymologists and so on) dispute it. There are treatment. other explanations offered, but ultimately the origins of the word are not known for certain. At There is a third possibility, of course, and any rate the acronym hypothesis looks like that is that the best explanation is a being a myth, sadly. But it serves as a useful combination of these factors. Jointly they may warning. The port-out-starboard-home be more plausible than either one on its own. explanation is so plausible, and so pleasing, that once people have heard it, they want it to be When in doubt true; and they are disappointed when they find out that it is at best dubious, or worse still false. It is often quite easy to think up a plausible explanation, or combination of plausible The danger of believing what we want to explanations, for some observed fact. But it is believe is a serious one. It is also one of the often very difficult to come to a confident reasons why critical thinking is so essential to decision as to the best explanation. Moreover, serious inquiry and the acquisition of as we have just seen, even the most plausible knowledge. explanation, which seems to tick all the boxes, 4.2 Explanation 141

Summary • The best explanations are those that are simple and that explain the most (have • Explanations differ from arguments, the widest ‘scope’). despite resemblances. • Even the best explanations may be wrong: • There may be many possible explanations they are, strictly speaking, hypotheses. for an outcome or event, though some are more plausible than others. End-of-chapter assignments 1 Which of the following short passages are was weightless. But in all three arguments, and which are explanations? locations the amount of matter remains the same, and this constant A Icebergs are formed from glaciers amount is what is meant by its mass. breaking off into huge chunks when D He did it for sure, because no one they reach the sea. The process is else had the opportunity, the motive known as ‘calving’. The glacier is or the training to do such a thing. formed from snow, so it consists of E He did it because he needed the freshwater ice. The oceans consist money and because an opportunity of brine (salt water), which has a came his way. significantly lower freezing point than fresh water. Therefore the sea 2 Study the following information and then around icebergs remains in a liquid answer the question that follows. state. Measurements were taken showing B Ice is less dense than liquid water. the growth of 16 fir trees planted at Consequently, ice forms on the the same time but at different surface of lakes and ponds, instead altitudes on a hillside. The results of sinking to the bottom. were recorded as shown in the graph. C In our ordinary everyday lives we use Height of tree (m) the word ‘weight’ as if it meant the 25 same as ‘mass’. For example, we ‘weigh’ cooking ingredients in the 0 500 kitchen to tell us how much to use, not Altitude (m above sea level) to measure how much downward force they exert on the scales. But there is a distinction, and in science-teaching it must be preserved and stressed. A bag of flour on the surface of the Earth has a different weight from the same bag on the moon: here it is approximately six times heavier. And in an orbiting spacecraft we would say it 142 Unit 4 Applied critical thinking

Which of the following, if true, would be a die. Leaving a climber to his fate, just to plausible explanation for the data recorded get to the summit of a mountain, was in the graph? (There may be more than one.) unthinkable in Hillary’s day. Then people climbed as members of large, A The fir trees planted at higher organised expeditions and knew each altitudes tended to be shorter. other as friends and colleagues. Not all of them were expecting, nor even B The higher up a hillside you go, the attempting, to reach the summit, poorer the soil tends to be. because it was the purpose of the expedition just to get one or two C Air temperature decreases with climbers to the top. It was a team effort, altitude. and the credit was shared. Once the mountain was beaten they could all go D The higher a tree is planted, the home, satisfied that they had achieved smaller its growth. their shared goal. Today Everest is besieged by swarms of individuals who 3 Re-examine the data in Doc 3 in have paid thousands of pounds for Chapter 4.1 (pages 131–2) and the their one chance to make it, personally, statistics discussed in this chapter from to the top. No wonder traditional Doc 4 (page 134). Suggest one or more mountaineering morals have been explanations for the widespread public thrown to the 80-mile-an-hour winds. perception (shown by the charts) that claims for injury are up, when official Is this passage an argument or an statistics suggest that they are stable or explanation? (Give a reason or reasons for falling. Which is the best explanation, in your interpretation.) your view, and why? • If it is an argument, identify its conclusion and summarise the reasoning. 4 (Harder task) Study the following short • If it is an explanation, state what article and complete the task below. is being explained, and what the explanation is. In 1953, New Zealander Edmund Hillary and the Nepalese Sherpa Answers and comments are on page 324. Tenzing Norgay became the first climbers to reach the summit of Mount Everest and survive. That was then. Now Sir Edmund has come out in forthright criticism of some 40 climbers who passed a dying man on the upper slopes of Everest, and left him there to 4.2 Explanation 143

4.3 Evidence Practically anything can be evidence: a summarising the figures and connecting the footprint, a bloodstain, a written or spoken information from Charts 1 and 2. statement, a statistic, a chance remark, an email, some CCTV footage . . . the list could Similarly, the figures in Doc 4 in Chapter 4.1 run to pages. (page 134) are evidence for the claim that: There is good and bad evidence, just as there [B] Overall, more than 3000 fewer claims are good and bad reasons (for a conclusion). were notified in 2007/8 than in 2000/1. Judging whether or not a piece of evidence is ‘good’ depends on what it is being used as These two claims between them could then be evidence for. There is nothing good or bad used to argue that, for example: about a percentage of people saying they think that false claims for personal injury are on the [C] The public perception of a dishonest increase. That is just raw data; a fact. It becomes ‘compensation culture’ is completely evidence when it is used as a reason for some mistaken. conclusion or verdict; or, to put it another way, when something is inferred from it. Expressed as an argument: From this we can see that ‘evidence’ and [1] According to a report by the UK House ‘reason’ have some overlap in meaning. of Commons Constitutional Affairs However, there are subtle differences in the Committee the vast majority of British way we use the terms in connection with people believe that more arguments. Recall once more the evidence compensation payments are being underlying the discussions in the previous two made than previously. However, the chapters. The charts in Doc 3 in Chapter 4.1 Compensation Recovery Unit reported (pages 131–2) could be cited as evidence for the that over 3000 fewer claims were claim that: made in 2007/8 than in 2000/1. The widespread perception among the [A] The vast majority of people believe that British public that there is a growing, more compensation payments are being and increasingly dishonest, made than previously, and that more compensation culture is completely false claims are being made. mistaken. But we could just as well say that the data in Whether we want to call [A] and [B] ‘evidence’ the charts gave reasons for inferring [A]. It is for [C] or ‘reasons’ for [C] is a matter of useful therefore to think of the numbers and preference. They are evidence because they are percentages reflected in the charts as raw factual and statistical; they are reasons because evidence (or raw data) which has been they are used in support of a conclusion. The extracted and processed into the statement, distinction is maintained when we say that [A]. [A] expresses the evidence in a form that the reasons (or premises) in [1] are based on the could be used as a reason (or premise) in an evidence provided by two sources. If [A] and argument. It also interprets the data, by [B] are warranted by the evidence from those sources, then [1] is well founded, and 144 Unit 4 Applied critical thinking