Principal Examiner Feedback

Summer 2014

GCSE Mathematics (Linear) 1MA0

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GCSE Mathematics 1MA0Principal Examiner Feedback – Foundation Paper 1

Introduction

The paper was accessible to all candidates, with all questions attempted by a good proportion of candidates.

The standard of basic arithmetic seen was often poor and marks were lost when candidates could not perform simple calculations, even though they understood the mathematical concepts being examined in the questions. Their comprehension of mathematics often out stripped their numerical ability.

It was noticeable that more candidates appeared to be showing their working out. This often benefited them as they faltered on the actual calculations.

The standard of algebraic manipulation was often below the required standard of GCSE.

Report on individual questions

Question 1

This was an accessible question for most candidates. It allowed candidates a positive start to the paper. Part (a) and (b) were usually correctly answered. However in part (c), the most successful strategy seen was to write all the numbers out to 3 decimal places and then compare. A common incorrect answer was to place 0.63 before 0.603 or misplace 0.6.

Question 2

This question was also well answered. The majority of candidates were able to give the 3 correct answers.

Question 3

Candidates were able to access this question. The concept of addition and difference were attempted by the majority of candidates. All too often full marks were missed because of incorrect arithmetic. Most candidates preferred to add the two sets of three numbers and then subtract. Most errors were in the additions attempted.

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Question 4

Part (a) was answered usually correct.

Part (b) was well answered, very few blank or incorrect responses were seen.

In Part (c), whilst only a minimal reason was required, many lacked clarity. Common incorrect responses seen were “it goes up in 4s”, “not in the four times table”, “it’s not in the pattern”. Correct answers usually referred to the sequence consisting of odd numbers or the fact that 372 was even, or both points and some candidates correctly used the nth term. However, a few candidates did confuse the terms odd and even.

Question 5

Part (a) was well answered, very few blank or incorrect responses were seen.

Part (b) was a challenging question for many candidates. The use of negative temperatures was ignored by some candidates who chose to work with absolute values. Others attempted to add the numbers given, but arrived at a variety of answers. The most able showed a full method and this was required for the method mark. The accuracy mark was only awarded on a minority of occasions.

A common error was to see 7 ÷ 7 = 0. It was also apparent in this question that candidates attempted to sum the numbers in their head. Unfortunately, when they got the wrong answer no marks could be awarded as there was no evidence where their total had come from.

Question 6

In Part (a), correct time was usually given.

In Part (b), a common incorrect answer of 11 was often seen. Showing candidates lack the comprehension to understand this question.

Part (c) was often more successful than part (b) with the correct time interval usually stated without any visible working out. The most common incorrect answer was 63 where candidates saw the hour difference but added 3 instead of subtracting it.

Question 7

Part (a) was well answered, with some answers seen on the scales. Many candidates found a quarter of 48 and 48 ÷ 2 ÷ 2 was often seen as working out.

Part (b) was more variable with some candidates able to read the scale, some able to divide by 3 and others able to do both. Marks were lost through poor arithmetic again in this question or because the scale was read as 650. Although candidates realised they needed to divide by 3 many tried to half and half again, obviously this did not arrive at the correct answer.

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Question 8

Part (a) was well answered with the occasional incorrect answer of ‘hexagon’ seen.

Part (b) was well answered. The majority of candidates were able to identify a pair of parallel lines. Part (c) was well answered. Most candidates could recognise an acute angle.

In part (d) a few fully correct answers were seen, with the majority of candidates scoring part marks on this question from either 10 or the independent cm². The perimeter was often confused with area when candidates counted the diagonals as 1cm. to give an answer. A significant number of candidates attempted to extend the shape, usually leading to an incorrect answer, rather than counting the squares.

Question 9

The majority of candidates scored 3 marks as all was correct apart from the missing label on the vertical axis. It was most common to see a bar chart, less frequent was a frequency polygon. Uneven bar widths were seen occasionally and heights were generally correct.

A key was usually given and the months usually written out in full or written so as to be easily distinguishable. Where some fell down was that linear scale was incorrect or written within squares.

Labelling the y axis was problematic for many and incorrect labels used included “temperature” or “frequency” or occasionally “y”.

Question 10

In part (a), the approximate rule was generally applied correctly to give an answer of 70. However, the exact rule although applied correctly rarely gave the correct answer due to candidates’ inability to multiply 20 by 1.8. This led to many different answers of which 21.6 appeared to be the most common. They were able, however, to gain 3 marks out of 4 by a correct difference being given to their subtraction as long as all the working out could be followed.

Part (b) was generally answered well, though it appeared that some used a simple form of ‘trial and improvement’ to arrive at their answer rather than a using inverse operations. Those who used the reverse operations sometimes incorrectly divided 110 by 2 first then subtracted 30 to give 25. A very small percentage of candidates used the exact rule and were able to score some marks from the special case consideration.

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Question 11

In part (a), most could count correctly and then give the answer as a fraction. For those that

went on to simplify this fraction, many did so correctly. Several just gave as the final

answer. The most common error was to see an answer of where candidates had counted the

unshaded squares instead of the shaded squares. Some did try to “simplify” further and so

scored only one mark. A sizeable minority gave the answer .

In part (b), the correct answer of 2 was often seen and the most popular incorrect answer was 1.

Part (c) was generally well done but it is disappointing to see that a significant number of candidates did not even attempt this question. Quite a large percentage of candidates got full marks for this question.

Of those who didn’t, common mistakes were splitting the shape in half and trying to shade sufficient squares to make one side a rotation of the other or shading squares correctly in rows 2 and 4 but leaving the top and bottom rows untouched.

Question 12

Part (a) was well answered with most candidates writing 3ac occasionally ac3 was seen.

In part (b), p³ or p3 were the common answers unsurprisingly seen. The size of the 3 was rarely debateable and the candidates’ intention clearly communicated.

Part (c) was not well answered. Many candidates scored 1 mark but –y was far too often seen as the second part. Occasionally 8x7y was written as a product.

Question 13

Part (a) was very well answered with the use of the word ‘and’ condoned.

In part (b), a high percentage of candidates gained full marks on this question. Those who missed out on full marks often gained one mark for one correct dimension. The rectangle was drawn in different orientations but this was acceptable.

In part (c), the full correct answer was frequently seen. However, actual measurements recorded on the given diagram were rare, as was the comment that it was a size 1 advert. Most went straight to £6.50 and multiplied by two giving the correct answer. The most common incorrect answer was £27.00, where the candidates has been inaccurate in their measuring, or had rounded the length to 45mm, evidence of the mistake was rarely seen.

Part (d) was a very well answered question on the whole – most candidates either got the question completely right or started off correctly by attempting to multiply £13.50 by 8, but then went wrong in their calculation.

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Common calculation errors were: multiplying 8 by £13 but forgetting to multiply 8 by 50p as well, using a “doubling” method but going one step two far (13.50 × 2 = 27 × 2 = 54 × 2 = 108 × 2 = 216), using a basic addition or subtraction method but with either using too few or too many 13.50’s.

A few candidates misread the question and multiplied the 8 weeks by an incorrect account.

Some students gave excellent conclusions as to why £100 was not enough, but quite a few just did the calculation, arrived at £108, and finished there. As this was not a QWC question this was not a problem, but candidates should always be encouraged to write some form of short conclusion to these types of questions.

Question 14

In part (a), there was much uncertainty whether to divide 3 by 5 or the other way round.

Many chose to do 5÷3 as this was possibly thought to be easier. However, if a conversion into pence was made the answer of 60p was much easier to calculate. Poor arithmetic was seen in this question and also money notation was weak, e.g. 0.60(p) and 06(p). There was a lack of awareness of the reasonableness of the answer as many incorrect answers if checked were obviously wrong.

In part (b), many candidates wrote down a first step but then found 100 ÷ 80 too difficult a calculation for many; 1.20, 1.24 were common wrong answers. 100 ÷ 5 as a starting approach was commonly followed by a second step of 20 ÷ 80 but then this proved too difficult to execute or became 80 ÷ 20 = 4 and £4 was often given as the final answer. Again a quick check would show this was obviously incorrect.

0.25p was also given as the final answer, thus showing a lack of understanding of notation. In this question correct notation was expected for the final accuracy mark.

Question 15

Candidates were often able to find the volume of one of the cuboids. They were able to see the need to divide the two volumes but the number of zeros present in the calculations was variable. 2 or 2000 were popular incorrect answers. Again the level of arithmetic restricted scoring on this question.

Another popular method was to find out how many small boxes could be placed inside the larger box but even with these three numbers wrong answers were still given. Some candidates could not successfully multiply 4, 5 and 10 whilst others chose to add them.

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Question 16

Only a few fully correct answers were seen because reasons, containing all the key elements, were rare. When reasons were given they were seldom all given. If attempted, angles in a triangle add up to 180˚ and angles on a straight line add up to 180˚ were generally correct, however ‘isosceles triangle equal to 25 since 2 parallel sides’ was the most common quote for the rarely mentioned isosceles triangle.

A common method used was to start with the large triangle to give 25 + 70 = 95 then 180 –95 = 85 unfortunately they then said x = 85 so no marks could be awarded.

It was rare to see “angle ADB=25” written down but 25 was seen labelled in the diagram and this received 1 mark.

Question 17

This question was not well answered and was not even attempted by a good number of candidates. Many who did attempt this question had more than 3 sectors so could not gain any marks others just used the given values directly as degrees. Some candidates did manage to draw one angle correct but it was doubtful how they did this without any evidence of working out. Freehand lines were also in evidence and candidates need to be reminded that this can often lose marks as part of a freehand line could fall outside the accepted tolerance.

If all three angles were drawn they were usually correctly labelled, however, a small number of candidates showed all their working, drew correct angles and then placed the labels in the wrong sectors.

Question 18

In part (a), many candidates ignored the fact that “5” had to be negative whilst others used 3 + –5 instead of 3 × 5. 15 + 8 was frequently seen but then it was either not completed or given as ± 23. Occasionally 15x + 8y was given as the final answer or even 7xy.

In part (b)(i) was not well answered and common wrong answers were: p = 10, p = 10p, p + 10.

In (ii) a follow through was permitted only if (i) was algebraic and many candidates scored one mark this way. A common error was to give 10p correctly in (i) but wrongly give 3p in part (ii)

Question 19

Candidates appear to find arithmetic with fractions difficult, all too often or = 1 were

given as the final answer. Even when candidates were able to give 15 as the lowest common multiple of 5 and 3 they could not go on to find the correctly associated numerators.

Some candidates used the grid method to find the answer, this worked for some candidates but others could fill in the boxes and then did not provide a final answer.

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Question 20

Candidates showed a variety of approaches to this question. Some used a two way table and filled in the gaps. Calculations were rarely shown in this case and sometimes simple mistakes were seen. The answer must be identified within the table to gain full marks.

Most candidates, at this level, did not use a two way table, they used a string of calculations using the numbers in the question. Some of these calculations were not sensible for example adding 21 and 18 others just added all the numbers given.

More successfully candidates often scored a mark form either finding 29 males or for realising 8 – 6 = 2 is the number of females who play squash. A second step was less often seen, their first answer needed to be used to find a second value which could go in the table, often this step was incomplete. Some candidates did manage to do 29 – 9 or 29 – 6 but not always 29 – 6 – 9.

Whilst 14 and 15 may have been given by candidates these were not always for the correct classification and when a lot of different calculations are possible centres should encourage candidates to clearly show what they are working out.

Question 21

Most candidates attempted this question, with very few leaving it completely blank, but the inability of candidates to deal correctly with fractions and percentages of amounts of money was highlighted quite starkly here.

Part marks were often scored. Most candidates could find one of the prices, however, a lot of

mistakes were made. For example off is not the same as 30% off and too many candidates

found 60% but then did not subtract this from the full price even though they had correctly

found off. Many wrote of 24 was 6, possibly coming from 24 ÷ 2 ÷ 2. Some found 60% of

either £12 or £24. Many successfully found 10% and multiplied by 6, or added 10% to 50%, but some found 50% then found 10% of their “50%”, technically calculating either 55% or 45%.

Even with one correct calculation achieved far too many candidates went on to add the discounts together, or a mixture of discounts and discounted tickets, or only added one child, or even adding two adults and one child.

Part marks were the modal score for this question.

Question 22

This question was attempted by most candidates. However, some wrote questions which were not related to the required topic. Candidates must consider books bought not read or preferred. Centres should ensure that candidates give exhaustive options, often the zero option was missing and always give a time frame. There are still some issues with overlapping options but it is pleasing to see that, at this level, hardly any candidates used

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inequalities, which are not acceptable in questionnaires.

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Question 23

In part (a), there was only one mark for this question and so both terms were required. This happened sometimes but often 2m² was correct and 6m was incorrectly given as 6 or 3m or 5m. Occasionally 2m² + 6m = 8m or 8m² or 8m³ was seen, there is no ‘ISW’ on algebra questions and so these answers did not score the mark.

Part (b) was not well answered. It appeared to be beyond most candidates. Occasionally 3xy was identified as a factor but the other factor was rarely seen.

Question 24

There were quite a few fully correct answers given. However, a significant number found the perimeter of the shape instead of the area. The concept of finding an area for a ‘pig’ did not seem to be an issue but finding the area of a simple compound shape was. Many failed to work out the hidden dimensions correctly, showing no working for any answer obtained. Those who found areas often included an overlap section, usually 16 × 6 with 7 × 10 or just considered 16 × 10 failing to subtract 4 × 9. The most popular correct area seen was 7 × 10 = 70. Some arithmetic errors were seen when calculating areas. Many were able to gain the first method mark but then far too many scored zero on second method mark.

The need to divide by 36 was understood by many but it was a challenge to actually carry out a suitable calculation. The most successful way was to repeatedly add 36 and get to 108 and even 144 then realising that this meant that the correct area of 124 could hold 3 pigs.

Question 25

This was an accessible question for many candidates. A good proportion scored 1 mark by rotating the shape through 180° but not always about the correct centre. Many correct answers were seen.

Very few candidates changed the size of the shape but some did draw a reflection.

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Question 26

This is quite a standard question but many pupils just left it blank. Some stated they were running out of time and so this may have accounted for why many were blank.

When candidates did attempt the questions the standard of arithmetic was appalling. Some found the difference between the two prices to be 60p, how could this be when one price ends in a 4 and the other in an 8? Others discussed the size of the bottle and whether it would fit in the fridge without doing any calculations at all, others just wrote down a size. As a QWC question both working and a statement was required.

A comparison of equivalent numbers of pints was expected to justify the answer but often 1.18 × 4 and 1.74 × 6 was compared or 8 pints and 12 pints by doubling, the candidates stated this was what they were doing and so showed a total lack of understanding of the required strategy.

There was poor evaluation with £1.18 ÷ 4 = 29.2 often seen and 1.74 ÷ 6 = 1.74 ÷ 2 ÷ 2 ÷ 2 was frequently stated. There was an over reliance of halving by many candidates. A simple but effective successful strategy was to find the price of 2 pints from the 4 pint bottle and multiply this by 3 to give a 6 pint comparison, £1.77 was often correctly given.

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GCSE Mathematics 1MA0Principal Examiner Feedback – Foundation Paper 2

Introduction

This paper was found to be reasonably straight forward at the start but a number of the later questions that caused some candidates problems particularly 18(b), 21, 24, 28 and 29. The paper produced a good range of marks for the award of grades. Errors were often made where the candidates did not read the question carefully e.g. question 3.

Generally speaking, the standard of straightforward algebraic knowledge was not very good as candidates tended to use trial and improvement methods in the solution of any equation. Unless a trial and improvement method leads to a correct answer then no marks are awarded unless trial and improvement is the focus of the question. Candidates usually gained more marks for using an algebraic approach to the solution in those questions where an algebraic method could have been used rather than by using a trial and improvement method. This was particularly the case in questions 26(a) and 28.

A significant number of marks were lost where candidates did not write down a statement of the result in the starred questions. Circling an answer is insufficient as we need to see a statement giving the required decision based on written evidence. A statement of how to work something out will also not gain any marks when a question requires an explanation.

It is still surprising to see the number of candidates who did not have access to equipment as many candidates chose to draw their answers to questions involving the drawing of straight lines freehand. This was not penalised where those lines were reasonably straight. Since this was a calculator paper it was surprising to see candidates losing marks for inaccurate arithmetic – presumably either because they did not have a calculator or because they chose not to use the one they had.

Some students give more than one method and more than one answer. If they choose one of the answers to write on their answer line that is the method which will be awarded marks, but many do not do so and give us a choice of answers. This was often apparent in questions 18(b), 21 and 24.

Report on individual questions

Question 1

A well understood question with all parts well answered by all candidates.

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Question 2

This question was well attempted by those candidates that had a ruler. A surprising number did not so were at a disadvantage in part (a) though they could find the midpoint of the line in part (b). The other part involving drawing was well answered though many drew a diameter rather than a radius with the occasional chord sometimes seen and sometimes the line extended beyond the circumference. Measuring an angle was usually well done though the supplementary angle was sometimes given.

Question 3

The standard bill type question was very well answered with many candidates giving the correct answer. The most common error in this question was to omit the second fish and chips and the three other puddings; these candidates were awarded 2 marks for an answer of £26.15 which would be the correct answer from their incorrect assumption. A few did not use a calculator when subtracting and gave an incorrect answer of £3.75 or £4.25.

Question 4

Less than half of the candidates answered this question correctly with many giving the

answer of 3.2 or 3.3 and a significant number wrote 3.2 which also was not accepted.

Question 5

This question was well understood but sometimes not very well answered as candidates tended to write 600 rather than 28 600 in part (a) thousand rather than 20 000 in part (b). The success rate for part (c) though was much higher with almost all candidates obtaining the right answer. A significant number of candidates included a decimal point in the answer as a delimiter and this was not accepted for the mark in part (c).

Question 6

Candidates answered this question very well with two marks being obtained by most candidates. Some candidates included Janette as well and were not penalised whilst those candidates that included duplicate entries were. A minority of candidates thought this was a probability question with answers such as “…it is more likely that a boy is picked…”

Question 7

Finding the congruent shapes was well answered, but many candidates could not find the triangle that was enlargement of triangle A as C and D were often given as the most common wrong answers. Interestingly though, many candidates could give the correct scale factor as 2 even without the correct answer for part (b)(i) being given.

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Question 8

This question was not well understood or well answered. In part (a) many candidates gave the answer of 36 extra squares needed, ignoring the 9 given squares and many just gave 9 or 18 as their answer obviously not understanding each side of the new square was 6 cm. Those who actually drew a 6 by 6 square and counted the extra squares were usually more successful. In part (b) more candidates understood the question but struggled with establishing the pattern needed to get to the seventh term. The candidates that were most successful were those that drew pattern number 7 and counted their triangles, sometimes incorrectly, though many used a build-up method with partial success.

Question 9

Candidates were much better prepared for this type of question and they performed very well usually giving the correct answer of £4.80. Due to the way the question was worded candidates could also score full marks for an answer of 40p, the saving per month. Interestingly some candidates followed their calculator display and showed lack of thought for currency by writing £4.8; this was not penalised in this question though those that wrote £4.08 were.

Question 10

A fairly typical response for this question was to give the answer of 11 i.e. the candidates gave the middle value of the original list rather than ordering the list first. There were also those cases where the candidates gave the mode and some even calculated the mean. Many candidates chose to use 8 of the numbers instead of 9 and lost marks because of carelessness.

Question 11

This question was well understood and part (a) was well answered with almost all candidates able to draw a kite with rhombuses and squares condoned. In part (b) almost all candidates were able to draw a rectangle but few candidates were able to draw one with a perimeter of 14 and many drew rectangles with areas of 14 cm2. Almost all candidates gained one mark in (c) either for drawing at least one correct line of symmetry or for drawing two correct lines of symmetry and drawing additional incorrect lines.

Question 12

Candidates were less successful in part (a) than part (b) as there was much confusion in candidates minds between seats and tables in part (a) whilst in part (b) the scenario was more understandable to them and as a result their success rate was a lot better. Many also calculated that there would be £17 left even though this was not required. A few lost the C mark as they failed to answer the question with yes.

Question 13

This question was not really well understood by candidates and many could not cope with the complexities of the diagram. Many simply divided the length and width of the garden by 4 whilst others tried to calculate the area of a flower bed.

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Question 14

The most successful part of this question was part (a)(ii) as most candidates could cope with counting the number of faces in the triangular prism. Part (b) was answered reasonably well though candidates at this level often find interpreting 2_D representations of 3-D shapes quite difficult. The part candidates made the most errors on was part (a)(i), finding the number of edges on the shape even though all 9 of them were drawn in for them.

Question 15

This question was reasonably well answered with many candidates able to give correct responses for a factor of 6 and a multiple of 6 though inevitably many mixed the two up. In part (b) it was gratifying to see so many correct responses with many candidates able to state that 2 was a prime number. Many candidates chose to describe square number or stated that 12 was a prime number or said things such as “all even numbers are odd”

Question 16

Parts (a) and (b) in this question were well answered but when it came to giving the equation of the line, few candidates were able to write x = 3. Many gave two coordinates on the line and the usual incorrect response of y = 3 was the most common error seen in this part whilst in part (a) the point marked was often (–2, 3) or (–3, 2).

Question 17

This question was poorly answered with the exception of part (a). Many candidates in part (b) just added up the 0, 1, 2, 3, 4 and 5 to give the answer of 15 and then wrote down the answer of 2.5 in part (c) obviously making the classic mistake of dividing by the number of rows in the table.

Question 18

Almost all candidates were able to change £200 into rand successfully in part (a) but in part (b) they struggled to work to the degree of accuracy needed to gain all three marks but many were able to score one mark by giving one correct conversion from rand into pounds or pounds into rand and some were then able to put together enough conversions to get close to the required accuracy and score two marks for a complete method. The quality of the conclusions made by a significant number of candidates was not of a particularly good standard as the comments made about the items that can be purchased, did not always reflect the figures they had calculated.

Question 19

There was a mixed response to this question. Many candidates tried to set up a questionnaire or draw a graph and scored no marks as we needed to see a standard 3 column data collection sheet with type of transport, tally and frequency as labels for the columns. Each correct column gained a mark. A few candidates lost a mark as they had columns headed frequency and total rather than tally and frequency though total was condoned instead of frequency.

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Question 20

The mode for this question was 2 marks. Almost all candidates realised that they had to multiply 1500 by 8 and then by 60 for which they received the two marks. Unfortunately most candidates could not take this to the next stage by multiplying by 330 ml as they either divided by the 330 or if they multiplied they then did not divide by 1000 preferring to use 10 or 100 as the number of millilitres in a litre.

Question 21

This question differentiated very well as most candidates could make a start by writing two equivalent fractions but only the best could give the correct conclusion from three correct equivalent fractions, decimals or percentages. Some candidates tried to use a diagram but didn’t realise that their diagrams were not comparable because they hadn’t split them into the same number of sections. It was disappointing to see so many candidates rounding

prematurely thinking that is equivalent to 0.6 so losing marks.

Question 22

Travel graphs are usually well understood and this was the case here as far as the horizontal line was concerned but drawing a slant line of the correct gradient proved too difficult for most candidates.

Question 23

This question about powers proved a bit too difficult for many candidates. Part (a) was the best answered as candidates could use their calculators to work out the correct answer but after this candidates did struggle with p5 often being given as an incorrect answer for (b). Part (c) was usually better answered and in part (d) a few more gave the correct answer of 6.

Question 24

Candidates could usually make a start on this starred question and many candidates scored 1 mark usually by dividing the numbers of students by 12. They then usually became unstuck as they did not know whether to round up or down within the context of the scenario. Those that worked out the correct answers to the pupil teacher ratio then struggled to work correctly with the number of teachers needed and the number of students they could supervise. They did not think about what their answers showed and were unable to interpret them to give the correct answer. For example dividing by 15, 13, 14 and 12 showed how many students were assigned to each teacher, and so needed to be 12 or less, but many gave the answer for the days where this worked out at 12 or more. Candidates who multiplied the number of teachers on each of the days by 12, then compared them, were much more successful; however this approach was not the most commonly seen.

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Question 25

This scatter diagram question was well understood and well answered with most candidates able to score marks usually for plotting the points although some plotted (14, 6) instead of (13, 6) so losing the mark. The mark for giving the relationship as negative correlation was usually scored though those candidates that answered with simply “negative”, without the word correlation, were not awarded the mark. Part (c) was answered correctly by more than half the candidates but lines of best fit were seldom seen.

Question 26

The equation solving in this question certainly gave most candidates a problem as they did not know how to deal with the fractional answer less than 1. Few candidates gave the answer

correctly as preferring to give it incorrectly as 1.5. Candidates were also poor at showing

the steps algebraically, often showing just the arithmetic required and that scored no marks unless a fully correct answer was given though a few scored a mark for showing 3p = 2. Part (b) was better answered with many candidates gaining one mark as they either omitted the zero or included the –5 in the inequality.

Question 27

This question was often attempted but not very successfully. Candidates could not deal with substituting x = 0.7 correctly into (x + 1)2 as they often wrote it as 0.7 + 12; others thought that 2x should be written a 2 + 0.7 rather than 2 × 0.7. The most common incorrect answer was 1.0115 where candidates had not used brackets when dividing by (2 × 0.7).

Question 28

Candidates struggled with writing the lengths of the sides of the trapezium algebraically which made accessing this question difficult. Candidates could get some marks though for a numerical approach though many wasted time with exhaustive but fruitless trial and improvement attempts.

Question 29

Candidates understood they had to find the missing side AB in this right angled triangle but often just added the two sides of 32 and 24. Only about a third of candidates realised they had to square and add the lengths if the right angled triangle with many subtracting instead. In part (b) a lot of the candidates assumed they had to find the areas of the two mirrors rather than find the perimeter of the mirrors and so scored no marks. Very few candidates were able to give a fully correct solution to this question though partial credit was often earned for trying to find the circumference of the circle and the perimeter of the triangle. Those who did try to find the perimeter did not take account of the fact that the metal strip is sold in lengths of one metre when trying to find the cost. Most candidates did not associate part (a) with part (b).

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GCSE Mathematics 1MA0Principal Examiner Feedback – Higher Paper 1

Introduction

Many candidates had been coached well for the examination and were able to carry out standard techniques with accuracy. Answers to QWC (Quality of Written Communication) questions generally showed enough working to allow the award of communication marks.

The standard of arithmetic was very poor. This was manifest in the lack of basic techniques especially with the 4 rules. Specific examples are given in the reports on the individual questions given below. In many cases, the level of presentation and of organisation of working left a lot to be desired and made it difficult for the marker (and the candidate) to work out the logic of the response.

Report on individual questions

Question 1

Part (a) was found to be straightforward by the majority of the entry. Of the rest, there were some who first found a common denominator and then tried to multiply numerators together and denominators together, which, if correct, would gain the one mark available. More often the ‘common’ denominator was left as that and the numerators multiplied together. There were many cases of 2 × 1 = 3.

In part (b) candidates were expected to find a suitable common denominator (invariably 15 for those who knew what to do). There were a surprising number of candidates who

subtracted numerators and denominators to get, for example, or who found the correct

common denominator but did not change the numerators. A small number of candidates added instead of subtracted – they lost the accuracy mark.

Question 2

This was well answered. Very few candidates were unable to show they understood what they had to do. There was a substantial number who lost a mark because they omitted a value (usually in the twenty row). Presumably they had not counted the entries in the table and compared with the twenty numbers in the list at the top. Most candidates gave a sensible key, often without units, but some lost a mark because they wrote ‘children’ as the units, which is clearly wrong.

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Question 3

There were many correct answers to this question. Candidates who designed and completed a two-way table were generally successful in gaining all 4 marks. Others were less so as they often lacked the organising principle already built in to the table. They generally started off confidently by finding the number of males (29) or the number of females who play squash (2). Subsequent calculations were then often confused as candidates could not keep track of what it was they were actually working out. In particular, they wrote down their calculations without making it clear (e.g. 2GSq would have done) what they were actually finding. There were too many cases of 50 – 21 = 19 seen.

Question 4

From a functional maths point of view, very many candidates showed they cannot tackle such a task as this one successfully. Generally, finding one third of £24 was no problem, although a minority of candidates thought that finding 30% would do or gave the wrong value for 24 ÷ 3. Similarly, many candidates were successful in finding 60% of £12 or £24, usually by dividing by 10 and multiplying by 6. However, although the mathematical techniques were carried out competently, there was an enormous lack of attention on what to do with the two figures already calculated. A very common error at this stage was to work out the sum of the discounts. Less common, but still frequent, was to added the adult discounted price to the discount for the children. This lost half the marks for the question. The common parlance of ‘off of’ does not help students in this! A significant number worked out the cost for 1 adult and 1 child and many left the otherwise correct answer as £25.6.

Question 5

Candidates were often fully successful in answering this question. The vast majority remembered to put in a time frame and suitable sets of response boxes were very often seen. A common error was for one pair of response boxes to have a figure in common – for example 2 – 4 followed by 4 – 6 with all other boxes being correct. This lost a mark. Other candidates did not include a zero. A few candidates did not read the question carefully enough and wrote the question ‘How many books do you read each week?’. The use of inequality signs was thankfully rare.

Question 6

In part (a) too many candidates could not carry out this simple expansion correctly. There were many responses of the form 2m2 + 6 or worse.

Part (b) also proved to be a challenge for many candidates. A few candidates could carry out a correct partial factorisation. A common error was 3xy(xy – 2), presumably displaying a misunderstanding of the interpretation of x2y2 as against xy2. There were, of course, many candidates who gained both marks.

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Question 7

Most candidates realised that they were expected to display suitable working out and declare their answer in a clear form. The vast majority of candidates proceeded by working out the area of the L shaped field. This was generally done successfully by dividing the shape into 2 parts, calculating those areas and summing them. Area by subtraction was very rare. Thankfully there were few perimeters found on this paper. However, a common error was to ignore the overlap between the 16 by 6 and the 10 by 7 rectangles so getting an area of 166 m2. Once the area had been found, most candidates demonstrated in some form that they had to find how many times 36 goes into 124. This was sometimes done by division, but often by counting up in 36s until 108 was reached. Some candidates displayed their lack of arithmetical skills by failing to do this accurately – for example 36, 62, 98. Candidates who tried to draw out areas of 36m2 on the diagram were rarely successful.

Question 8

Very many candidates were able to get full marks on this question. Many others were able to score at least 1 mark – either by a suitable straight line at the right distance from the given shaded region or from the arc of a circle drawn correctly. Some candidates lost a mark because they did not draw a complete arc that met the rectangle.

Question 9

Candidates were often successful in both parts, although with greater success on part (a) than on part (b). A few candidates rotated the figure in part (a) through 180° about the wrong centre (for example, a bottom corner of the trapezium). A small minority just plotted the 4 vertices giving no indication about whether they understood that rotations preserve shape. Part (b) proved more of a challenge for weaker candidates. The two main errors were – the correct size and orientation, but not with O as the centre, and the correct centre but the wrong scale factor (usually 2, less often 4). Many candidates interpreted ‘centre of enlargement’ to mean that the bottom vertex of the enlarged triangle had to be anchored at the origin. Candidates who drew a shape 3 times the size but in the wrong orientation did not score any marks.

Question 10

Most candidates knew that they had to find the price of equal quantities of milk. This was often unit prices, but also the price of 2 pints or 6 pints or 12 pints. However, this was one of the questions in which many candidates displayed a woeful lack of numerical ability. One common and sensible way to tackle the problem was to work out unit prices for the 4 pint and the 6 pint containers. This involved working out 1.18 (or 118) ÷ 4 and 1.74 (or 174) ÷ 6. There were too many cases where the divisions were completely incorrect and many cases where candidates could not deal correctly with the case 1.18 ÷ 4. Commonly, the answer was given as 29.2 from the remainder of 2, rather than the correct 29.5. Another error was to divide 1.74 by 2 then by 2 then by 2 presumably in (mistaken) analogy to dividing by 4. There was also evidence of candidates being unable to multiply decimals – for example 1.18 × 6 or 1.74 × 4 were often done by repeated addition.

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Question 11

Candidates attempts generally fell into three groups.

(a) Those who worked out 360 ÷ 5 or 540 ÷ 5 and were able to identify that they were finding the exterior angle or interior angle respectively. They generally went on to score all 3 marks.

(b) Those who worked out 360 ÷ 5 or 540 ÷ 5 but were confused over which angle they had worked out – they generally scored 0 marks as the mark scheme was such that if it was clear they had confused interior and exterior, then they got 0 marks.

(c) Those who had little idea – too commonly thinking that the interior angles were 60° for example. They invariably scored 0 marks.

Once again, some candidates lost marks because of numerical weaknesses. In this question this was often an error of the form 360 ÷ 5 = 62, for example.

It was pleasing to see some candidates giving reasons at each stage of their calculation.

Question 12

In part (a) most candidates were able to substitute the given value correctly into the formula. After that, there were many problems as a result of weakness with basic numerical techniques – firstly some candidates tried to expand the brackets but quite often did this wrongly by multiplying the 5 by the 77 only and leaving 32. Others worked out 77 – 32 and got the wrong answer whilst others did get the right answer but could not multiply accurately by 5. In addition, for those that got the correct numerator of 225 many could not even begin to divide by 9. Those that did get to the correct answer of 25 almost invariably made the correct

conclusion. It was rare to see candidates who got to carry out the division first, so

simplifying the calculation.

In part (b), the most common error from those candidates who understood what they had to

do, was mismanagement of the 32 term with answers of the form often seen.

Candidates should write out every single step when rearranging a formula. The mark scheme is designed to reward those who show a sequence of logical, algebraically correct processes.

There were a few flow chart attempts – these had to be correct the F to C way and then display that the order of operations had to be reversed as well as each operation being replaced by its inverse. Full marks were only given when the flowchart was correct and translated back into a correct algebraic formula.

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Question 13

The key word in this question which very many candidates overlooked was ‘estimate’. Unless there was an approximation done somewhere in the process to get to the answer full marks could not be achieved. Many candidates tried to calculate will the full figures – their working tended to be confused and their presentation so disorganised that it was not possible for markers to follow it. Candidate attempts tended to fall into two groups:

(a) Calculate the number of seconds in one day (86400) and then divide by 2014 (or 2000).(b) Divide 2014 by 60 to find how many minutes there are between prizes (about 33) and

then either work out how many prizes roughly this meant per hour or divide the number of minutes in a day by 33 (or 30).

The first method was bedevilled by awful arithmetic - a common error being 60 × 60 = 1200 to start off with. Very few candidates started with the calculation 60 × 60 × 20 and went on to divide the answer by 2000 which is possibly the most direct way. It was disappointing to see candidates who clearly had a good grasp of what they were doing carry out such calculations as 86000 ÷ 2000 = 43000 (or 430 or 4300).

The second method generally worked well especially for those who realised that 33 minutes can be approximated by half an hour so a good approximation to the number of prizes is to double 24.

Question 14

Few candidates knew the correct conversion despite this being stated as required knowledge in the specification. Of those that knew the 5 miles = 8 km conversion, most could then carry out the rest of the calculation correctly to get full marks. A few impressive candidates knew that 50 mph was the same speed as 80 kph and were able to complete the question very succinctly. A few candidates did not use a sufficiently accurate conversion but still gained some of the marks. These candidates generally used 1 mile = 1.5 km. If they used this conversion correctly then they were awarded 2 marks for the question. Most candidates had no idea of the equivalence and either ignored the fact there were different units to get an answer of a little over 9 hours or made a conversion by multiplying by 10 or 100.

Once again, there was evidence of poor numerical skills with the division by 50 causing problems. There was, for example, little sign of cancelling the 0s or of doubling the 480 and the 50 when working out 480 ÷ 50 and very often an answer was attempted by using some sort of build up method.

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Question 15

The values of y corresponding to positive values of x were generally worked out correctly. There was less success with the negative values, especially the value of y at = −1.

In part (b) values were generally plotted accurately and the points joined with a smooth curve, although the occasional set of straight line segments was also seen.

Part (c) proved beyond most candidates. Correct solutions were split between those who connected up the whole question and drew the straight line with equation y = x + 3. They were then able to pick out the required values of x for the two marks. Other candidates restarted, rearranged the equation and solved it, usually by factorisation. If the two values of x were given then the marks were awarded. Some candidates spotted that x = 4 satisfies the original equation, but without any of the two approaches shown they did not score any marks.

Question 16

In part (a) candidates were expected to read off the values of the upper and lower quartiles from the box plot and then to subtract. The standard of subtraction was very poor with 5.6 – 4.85 often been worked out as 0.85. Even worse, it was sometimes worked out as 1.25. Of course, many candidates did not get that far and commonly worked out the range. Reading off the scale was also a challenge for many students.

Part (b) was a problem for those candidates who did not have a grasp of the meaning of the quartiles and that the upper quartile essentially divides off the upper 25% of the population. Some candidates had some idea but worked it out as the upper 75%.

In part (c), candidates only scored a mark if they referred to a meaningful statistic from both the distributions and made a comparison. For many candidates this comparison naturally involved the median. A second comparison had to come from a measure of dispersion in keeping with practice from previous examinations. Candidates could compare the range or the interquartile range. For full marks one of the comparisons had to be in context (rather than as an interpretation) so a reference to distance ran, for example, was expected. Many candidates were unable to abstract meaningful statistics from the box plots and resorted to vague answers such as ‘ they ran further in the first half than in the second half’ which, of course, scored 0 marks. Answers which just referred to maximum and/or minimum values were not awarded any marks.

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Question 17

Parts (a) and (b) were essentially knowledge based For part (b) a few candidates left their

answer as . This was not awarded the mark.

For part (c), candidates were expected to adopt one of two strategies. The first was to reduce each of the given numbers to an ordinary number and them compare sizes. If a candidate did the conversion correctly for at least 1 number, they were awarded the method mark. The second strategy, much more rarely seen, was to write each number in standard form. If a conversion was done correctly for at least one number then the method mark was awarded. Many candidates, however, did not show what they had done and went straight to writing down the 4 given numbers.

Question 18

This was a standard simultaneous equation question which was, for some candidates a single step to eliminate one of the variables. Most candidates who had an idea of what to do multiplied the first equation by 3 and added. Those that subtracted were not awarded any marks. Others multiplied the second equation by 4 and subtracted. Those that added were not awarded any marks. In fact, elimination rather than substitution was the overwhelmingly commonly seen approach. Often, the elimination was carried out incorrectly with the difference between 12x and – x being found as 11x, for example. Once again, arithmetical weakness meant that candidates were losing marks. Typical errors included:

Getting to 13x = 91 and failing to go any further. Working out the difference between 64 and 25 and getting 41.

It was a pleasure to see some candidates properly checking their solution.

Question 19

Candidates were expected to show how they could find the gradient of the given line by using a variant of rise ÷ run. Many candidates were unable to do this and had no idea of what a gradient is. Some candidates were able to give the correct gradient for the given line L1, but then gave a different coefficient of x for L2. Candidates were much more confident in assigning the value of −5 to c in y = mx + c.

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Question 20

This question was seen by candidates often successfully as one about similarity in context. Candidates were expected to find a suitable scale factor, for example, 1.5, or to do some work on equating ratios of corresponding sides. They had to write their equation in a form which enabled them to rearrange to find the unknown side if they did use ratios before they were awarded marks.

A few candidates realised that they could turn the sheet through a right angle with respect to the photo. This was accommodated in the mark scheme.

There were many attempts to equate areas in some form. These scored no marks unless there was a reference to the square of the scale factor, for example.

Question 21

There were a variety of methods to complete this problem with its complex configuration. The most common successful approach was to calculate the reflex angle BOD and the angle at the circumference BCD, then use the angle sum of a quadrilateral together with angle OBC = 15°. Other approaches were rare. They included using the alternate segment theorem (although often wrongly applied), or using angle BOC = 150° and angle BOD = 140° followed by using angles round the point O and a suitable isosceles triangle.

In many cases candidates wrote down figures but did not relate them to the angles found. In this case the marks could often not be awarded unless the 55o was given as the answer. Many candidates sensibly put values of angles on the diagram and these were accepted as evidence of correct processes.

Question 22

In part (a) candidates who had an inkling of what to do, generally scored at least 1 mark. The most common errors were shown with the coefficient with values 3, 9 and even 81 commonly seen as well as and other variants.

Part (b) proved to be a challenge despite the question being solely one of standard techniques – factorise both numerator and denominator and then cancel any common factors. In very many cases candidates did not do this and so scored 0 marks. For those candidates that spotted the obvious difference of two squares many sensibly used what they had found to help them find the factors of the denominator. Sometimes the common factor was misidentified as (x – 3) instead of (x + 3) and so gave the wrong factorisation as (2x +1)(x – 3). A few candidates spoilt their good work by trying to cancel their answer of

.

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Question 23

This was selection without replacement and many candidates did not appreciate this. Common responses were to put repeats of the first set of branches on the second set of branches. Some candidates used denominators of 8 on the second branches and a few had the correct fractions but on the wrong branches – typically on the bottom second pair.

For part (b), candidates were expected to use the probabilities they found in (a). Candidates were awarded a mark for identifying at least one correct case and multiplying appropriate probabilities. Commonly one of the three cases was left out. This was sometimes the Yellow/Yellow case where candidates may have misinterpreted ‘at least’ and sometimes one of the Red/Yellow cases. Again, there was some evidence of poor arithmetic, but less strong in that most candidates who knew what to do could also multiply fractions correctly.

A minority of candidates solved the problem by using the complementary event. These were generally successful.

In both cases some candidates wrote what would be correct answers for compound events at the end the second branches of their tree diagram. These were not acknowledged for part (b) unless they were clearly indicated by the candidate that they were to be used in part (b).

Question 24

Candidates who had some idea of how to find the vectors and in terms of m and n, generally scored at least two of the three marks. The third mark was to give a reason based on the forms for and of why the two lines are parallel. Generally candidates earned the final mark by stating that 2n – 2m was a multiple of n – m. In general, notation was poor, with arrows above vectors rarely shown and with underling of m and n usually absent.

Some candidates did not read the information carefully enough and found that and were half the values given in the answer. These candidates could score a maximum of two marks.

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Question 25

Part (a) was done correctly by those candidates who understood the standard process of rationalisation. Answers in any correct form, such as or were accepted for full marks. If candidates went on to attempt to simply their answer and gave a subsequent incorrect answer then they were not awarded the final A mark.

Some candidates think that they can rationalise the denominator of the fraction by squaring the top and squaring the bottom presumably under a misapprehension that they are dealing with equivalent fractions.

Part (b) required candidates to expand the square – in many cases this proved too much, with many cases of the equivalent of . The use of was rarely used even by successful candidates. Some could expand the brackets correctly, but could not see how to simplify their square roots so unsimplified answers such as were seen. Many went on to ‘simplify’ wrongly, giving answers such as .

Question 26

The first two parts of the question were basically about how well candidates knew their trigonometric curves. The response was very poor with very few being able to give the correct coordinates. Surprisingly for this target level, there were candidates who gave the correct values, but reversed – for example (0, 180) instead of the correct (180, 0).

The next part of the question was meant to assess how well candidates understood transformations when applied to the cosine curve. Again, correct answers were few and far between as most candidates did not seem to appreciate the basic structure of y = cos x as evidenced by the first part of the question with the sine curve so were unable to relate the transformed curve to the original one.

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GCSE Mathematics 1MA0Principal Examiner Feedback – Higher Paper 2

Introduction

Many candidates were able to gain marks on the unstructured questions, whilst still gaining marks on questions which had a more traditional style.

To gain the highest marks candidates had to demonstrate high order thinking skills in a range of questions, not just in those questions towards the second half of the paper.

This is a calculator paper. It was evident from some work that candidates were attempting the paper without the aid of a calculator. This is not advisable, since calculation errors will cost marks. Premature rounding continues to cause problems for some candidates in multistep problems.

The inclusion of working out to support answers remains an issue for many; but not only does working out need to be shown, it needs to be shown legibly, demonstrating the processes of calculation that are used. In not all cases were numerical figures clear.

Some notes for centres:

Be aware that in order to gain the highest grades proficiency must be shown across the whole paper, including the easier questions in the first half of the paper.

All candidates need to come to a calculator with a calculator. In using a calculator answers should always be written to full accuracy, and continue to

be used as such in multi-step problems, addressing requests for rounding only at the final stage, and after a completely accurate answer has been demonstrated.

The inclusion of working out to support answers continues to need emphasis at a time when the demand for working out for some questions is increasing. Candidates should also ensure they write their numerical figures to be legible.

Centres need to continue practicing the solutions to unstructured questions and multi-step questions, and QWC questions which require a statement in additional to presenting working to problems.

Report on individual questions

Question 1

This question was done well by a good number of candidates, however there were also a surprising number of incorrect answers. A common error which lost a mark was in giving the coordinate without the brackets. A small number of candidates listed the values between 2 and 6, and from 3 to 8 and “found” the midpoint by crossing off matching values from each end of their lists. For the most part, this was done successfully. The most common incorrect approach observed was to subtract the two coordinates and this gave an answer of (4, 5). A few candidates attempted to complete this question by labelling the axes despite the diagram being labelled as not to scale.

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Question 2

The majority of candidates were able to correctly plot the two points accurately in part (a), and in (b) describe the relationship between the temperature and the units of gas used, although a significant number described the correlation between the variables.

The best way of answering part (c) was to draw a line of best fit on the diagram, and then to use this to read off the required value. Very few drew a line, some candidates joined point to point, and some lines with a positive gradient were seen. Nevertheless candidates in most cases were able to state a reasonable figure to gain the marks.

Question 3

This question was very well done by the majority, clearly well prepared for this. Some candidates did not appreciate the order of operations on the calculator and failed to get the accuracy mark provided they showed the substitution in to the expression. The candidates that were not well prepared often split the expression when making the substitution and so did not gain marks if the answer was incorrect. A few students did not give the answer to sufficient decimal places, but they were very much in the minority.

Question 4

The vast majority of candidates were successful with this question. A small number used the area formula rather than the required formula for circumference. Centres are advised to remind candidates to show their working because many students gave an answer of outside the required range without any working and so lost both marks. Those candidates who did not give the answer to the required accuracy were able to obtain a method mark if they showed the correct process. Some squared the diameter and a very small number squared π.

Question 5

A good number of candidates were able to collect two marks here. Where candidates obtained one mark this was often due to giving translation as the transformation, but then describing the movement rather than giving the vector, giving an incorrect vector or writing the vector incorrectly as a coordinate. Common errors with the vector were incorrect signs on the two elements and transposition of the two numbers. It was pleasing to see that a relatively small number of candidates described a completely incorrect transformation, however there were a significant number who gave more than one transformation, despite the instruction in the question, and therefore lost marks.

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Question 6

This question was generally attempted with candidates usually scoring at least 1 mark but with many common errors. Differing units between times given in minutes and speed given in miles per hour being at the root of these errors.

A common error was to omit the 30mins rest time giving an incorrect answer of 1.45pm. Some candidates calculated the overall time correctly as 5 hours 15mins or 315mins but then gave this as their answer rather than working out the resultant time. All these errors could have been avoided by reading the question properly. Other errors relating to time included

those candidates who calculated but then used this as 4 mins or interpreted it as

40 mins (giving 10:19am or 10:55am), and weaker candidates who added the 20, 25 and 30, or failed at the final stage by writing 5.25 as 5 hours 25 mins. It was not speed that was the main challenge in this question, it was simple understanding and manipulation of time.

Question 7

Most candidates were able to achieve 2 marks for correctly calculating the angle as 54 degrees. However, many candidates were not awarded the mark for correct reasons for their chosen method. Frequently only one reason was offered, or the vocabulary used was ambiguous or not sufficiently rigorous for geometrical reasoning. It was not uncommon to see confusion between alternate and corresponding angles.

Question 8

In part (a) most candidates correctly identified 13, 11 and 5 as the relevant numbers and got

credit for adding these. Some then went on to give the answer as 29 or . Some allowance

was made for those misreading the question as greater than 5 people, but otherwise accurate

reading of the question and the table was required. A common incorrect answer was , from

those reading the question as the probability of 5 only. The calculation 11 + 13 + 5 was frequently evaluated to be 30 or 19.

In part (b) many correct answers were seen. Some calculated , some did 1500 × 13. Some

incorrectly gave their answer as . A small number of candidates mistook the word

estimate to mean find a rough answer rather than a calculation based on probabilities.

In part (c) the question was often well answered with candidates showing that they understood the problems with the sample. However some candidates were not explicit enough leaving their response open to interpretation. We were looking for reasons in context so no credit was given for responses such as “it’s biased”, “it’s not varied” or “it’s not random”. Also, the two answers needed to be relating to different themes (e.g., sample size, timing) so that “children will be at school” and “adults will be at work” would only gain one mark since both relate to the timing of the survey. Some candidates referred to the nature of the question (e.g. the question was too personal) which was not relevant.

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Question 9

This question was a good discriminator. Many candidates labelled AB correctly and were awarded 1 mark. However, a common mistake was to think DC was 4 times longer than AB instead of 4 cm longer, with 8x or x + 4 often seen on the diagram. Another common, though lesser seen problem appeared to be pupils becoming confused between the act of doubling a side and squaring it, leading to AB being labelled as x2. Those taking the algebraic route usually attempted to add the 4 sides together and could simplify their expression. It was disappointing to see that so many candidates could not put together an algebraic argument and resorted to Trial and Improvement, usually stopping at an x value of 5.665. It must also be noted that many candidates used decimals and not fractions, but did not appreciate the difference between terminating and recurring decimals. Candidates need to understand that a

recurring number is a perfectly acceptable answer and best left in fraction form or or 5 .

Rounding or truncating an answer does not always gain the accuracy mark.

Question 10

Parts (a) and (b) were usually well answers, the only common errors being the addition of indices in part (a). In part (c) most candidates earned the mark, but some failed to subtract 3 correctly from 9, or divided it. An answer of 26 was accepted. It was disappointing how many candidates were unable to see the way to finding the answer. Many attempted trial & improvement approaches, whilst for many it was knowing what to do with the 64, resulting in many divisions by 3, or failed attempts to find the cubed root on the calculator.

Question 11

This question proved good at differentiating the candidates with a range of marks being awarded. The first method from the mark scheme was definitely the more popular approach to this question. There were a good number of fully correct answers, although it was a little disappointing to observe the number of these which did not include the 0 in 186.20 which was not penalised on this occasion. The most common error amongst otherwise good responses was when inconsistent units were used to calculate the volume.

Where part marks were awarded there were a variety of different reasons for this. Common reasons for a mark of 4 were not rounding to 7 bags and making an error in an otherwise correct calculation at some stage. Marks of two and three were also commonly given. Three marks were often awarded for having found a volume using inconsistent units followed by carrying out the remainder of the calculation correctly. Two marks was a frequent score for calculating the cost of the gravel before discount for some number of bags and then calculating the discount correctly, this often followed on from calculations which gave areas rather than a volume.

A reasonable number of candidates lost a mark as they found 30% and then did not subtract from the original amount to obtain 70%. Despite this being the calculator paper, many candidates used a non-calculator approach to find the percentage. A disappointing number of

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candidates did not recognise the need for consistent units to calculate the volume or, where they did, were not able to correctly convert the depth of the driveway to metres.

Question 12

In part (a) most candidates were able to gain B1 by recognising that the sequence increased by adding 5 each time and hence writing 5n, but there was a significant number writing 5n + 4 or even n + 5. Candidates need to be encouraged to check their rule to see if it works for the next term in the sequence

In part (b) responses were generally poor. A small number of candidates tried to find the nth term from first principles, with little success. Those who did realise that the two sequences were linked, often failed to double both terms resulting in the expression 6n – n², which was the most commonly seen incorrect response.

Question 13

Part (a) was usually attempted with full marks often awarded. The majority of candidates understood that integer values were required. There were a large number of candidates who either included by 0 and 5 or excluded by 0 and 5, possibly due to an uncertainty in the difference between inclusive and exclusive inequalities.

Part (b) was generally answered well, with candidates reaching a solution of 4.5 and scoring at least one mark. Many candidates continue to replace the inequality sign with a equals sign for solving with too many failing to return to the inequality sign for their final answer or just giving ‘4.5’ and so losing the accuracy mark. The majority of candidates who scored full makes carried out correct algebraic manipulations using inequalities throughout. The most common errors were those who multiplied out correctly (6x – 12) but then made a mistake with their algebraic manipulation, e.g. 6x >3; multiplied out incorrectly, e.g. to get 6x – 2; or multiplied out correctly (6x – 12) but then left their answer as 6x – 12>15 or 6x > 27.

Question 14

Almost all candidates attempted this question and almost all of those who did achieved at least one mark. This was generally for multiplying their number of boxes and packs by the correct price and totalling the cost. However, too many candidates were unable to find the first common multiple beyond 60, possibly as a result of not reading the question carefully. Those candidates who listed multiples and then used 96 or 120 rather than 72 were able to access some of the marks. Methods were sometimes confused, but examiners were able to credit sound working where this was shown. Again this highlights the importance of showing working.

Curiously, some candidates inferred from the word “least” that the question involved finding lower bounds. Where there was correct method shown again some lost valuable marks due to incorrect processing – seemingly not having access to a calculator. Most students however did achieve the final method mark.

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The correct answer on the answer line was often left as £25.8 without the zero which although wasn’t penalised here is not good practice when dealing with money notation.

Question 15

In part (a) most candidates identified Pythagoras as the best way forward in this question and ended up with the correct answer. Some attempted to use trigonometry and this was generally unsuccessful. Many showed no working.

In part (b) few candidates were completely successful on this part. Many attempted to use trig ratios, although these were often incorrect. Those using the sine or cosine rule together with their answer from (a) were less successful. Some assumed the triangle was isosceles. Some seemed not to realise that one of the triangle angles was needed. Many did not appreciate which angle was needed and, since the angles were often not labelled, it was not always clear to which angle they were referring. It was clear that many students did not know how to find a bearing, with angles being subtracted from 180 or the acute angle being given. Since the question asked for a calculation of the bearing, measuring the angle gained no credit.

Question 16

There was usually some evidence of the correct calculation being performed, but frequent errors in writing the answer correctly as required. The most common error was in writing the answer as 18.75 x 107. A few candidates attempted to add the given numbers rather than multiply.

Question 17

Candidates generally scored full marks or no marks. Those who were successful usually

worked out the scale factor, preferring to express this as 1.5 rather than . Some did go on to

use this incorrectly in part (b), multiplying by 1.5 instead of dividing. Using ratios of sides was rarely seen.

The most common error was to view the relationship as one involving addition and subtraction rather than a multiplicative relationship: the most frequent incorrect answers were LP = 8 and BC = 9. These results obtained from 9 – 6 = 3, and 5 + 3 = 8 and then for the second part 12 – 3 = 9.

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Question 18

In part (a) there were many different attempts at working out the answer. The most common error was to calculate simple interest for the second year as 1.5% of 200, rather than compound interest 1.5% of 206.60. A significant number of candidates scored M1 for £206.60 then made this error. This led to a common incorrect answer of 209.60. Less common errors included calculating a 4.8% increase on 200 (adding 3.3 and 1.5), and using

1.33 and 1.15, instead of 1.033 and 1.015, as multipliers or using % as equivalent to 3.3.

Most candidates rounded to the nearest penny, with some failing to do so and giving £209.699. Too many candidates continue to find percentages using non-calculator “stepped” methods.

Part (b) was rarely completed successfully. A large number of candidates attempted to subtract 12.5% from 225 and 5% from 535.50 to find the original amounts, with others adding 12.5% on to 225 and 5% to 535.50. Whether this is due to a lack of knowledge of the required method or an inability to understand the question is unclear. Those candidates who did appreciate that the original amounts were 112.5% and 105% usually went on to gain full marks, with some failing to gain the C mark as values were simply stated and no comparison given.

Question 19

This question was not attempted by all candidates. Many attempting the question did not give co-ordinates in the correct order; usually Y and Z being reversed. Candidates using the

correct method to determine the co-ordinate often lost the accuracy mark due to giving 1 as

1.3 instead of 1.33 … Many candidates failed to interpret the ratio 1:2 as meaning the line

was divided into and ; often the coordinate of P was divided by 2 instead of 3, possibly

from candidates assuming 1:2 as equivalent to .

Question 20

There were relatively few good answers to this question. The most frequently seen incorrect method was 72 – 67.8 = 4.2 followed by 4.2 + 72 = 76.2. Candidates need to practice mean in a variety of situations rather than just rote learning of calculation of a mean from a total divided by frequency. It is important to know that there are three elements in mean calculations and reversing to find totals is essential. Those who gained partial marks usually found either 3960 or 1695 but then couldn’t see how to complete the method, sometimes dividing by 55 instead of by 30.

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Question 21

In part (a) most candidates were able to score at least 1 of the 2 marks. The most common errors were to combine –5y and –2y as either 3y or +7y or to give –2 × –5 as –10. Confusion of multiplication and addition of directed numbers was apparent for a few candidates.

Part (b) was less well done. A large proportion of the candidates made no attempt at this part of the question. The fact that this was to be proved algebraically cannot be over-emphasised. A higher level candidate should be able to square a bracket and simplify an expression. The statement to prove has to be based on the algebraic factorisation showing that it was a multiple of 2 or 2n and hence must be even. Too many candidates substituted numbers rather than attempting an algebraic proof. Those candidates who did attempt to use algebra sometimes made errors in the expansion of (2n + 1)² or forgot that –(2n + 1) gives –2n – 1. A small number successfully obtained 2n(2n + 1) and gave a convincing explanation that the expression was a multiple of 2 and so even.

Question 22

It was noticeable that those who had chosen to draw a probability tree diagram were much more likely to get the answer correct than those who hadn’t. The majority of candidates showed some understanding of probability and were awarded 1 mark for P (tails) = 0.4 but many failed to cube “0.4” and multiplied by 3 instead, which highlighted a poor understanding of probability as their answer was greater than 1. A small number of candidates did not recognise that 0.064 was less than 0.1 and so came to an incorrect conclusion. There were many fully correct responses.

Question 23

In part (a) the vast majority of candidates were unable to communicate that a stratified sample is proportional in its nature. Many thought the essence of a stratified sample is purely about subdividing the population into groups. Common incorrect answers referred to taking equal amounts from each group or describing random or systematic sampling techniques. The quality of written communication was poor and candidates clearly found it difficult to express their understanding.

In contrast part (b) was well answered. In the cases where candidates only gained 1 mark this was either because they left their answer as 26.2 or rounded up to 27 people. Of those candidates who failed to score many of them often used the correct 3 values but reversed the order of division or multiplication.

Question 24

The responses to this question were mostly awarded full marks or no marks. A common misconception was to read the question as a direct proportion problem with many candidates giving 18 as an answer from (12 ÷ 4) × 6. Of those who started with the correct relationship most went on to achieve a correct answer although there were a significant number of candidates who failed to rearrange the formula correctly to find k. Those candidates who approached the problem by a numerical route gained 1 mark for 6 ÷ 4 (1.5) but often used the 1.5 as a multiplier rather than a divisor.

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Question 25

Most candidates were able to identify a correct equation for one or both parts of the shape but there were many errors. The most common included failure to divide by 2 for the hemisphere, squaring instead of cubing the radius for the sphere and using 14 as the height of the cone. It was surprising how many candidates squared or cubed pi. Some candidates did not seem to identify any equation from the formula sheet. Occasionally early rounding of an interim answer resulted in an inaccurate final answer.

Question 26

The algebra required to solve these simultaneous equations was beyond the capabilities of most students, although the majority of students attempted the question. The first step was to perform a substitution into the first equation. Those who did this were often able to go on to expand their squared bracket, although a frequent error occurred with the squared term. Many students were able to simplify their quadratic equation into a form to be solved either by factorisation or by the use of the quadratic formula. Many students stopped at this point. It was pleasing to see a few go on to solve the quadratic, and to realise that their values for x and y needed to be correctly paired.

Question 27

This question discriminated well, even amongst the most able candidates. Of those who were successful the most common start to solving this problem was using the Sine rule to find the angel at D. In cases where candidates failed to score any marks attempts at Pythagoras or trigonometry for right angled triangles were commonly seen, of those who did recognise the need for formulas for non-right angled triangles they proceeded to misapply the values to the

cosine rule or substitute the given values into the formula , showing a lack of

understanding of the included angle.

Some clearly able candidates worked out the required information using trigonometry but then thought the area of the parallelogram was found by multiplying the two side lengths together. Premature rounding lost the accuracy mark in some cases.

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GCSE Linear Mathematics 1MA0June 2014

1MA0 A* A B C D E F G1F Foundation tier Paper 1F 71 60 50 40 30

2F Foundation tier Paper 2F 70 60 51 42 331H Higher tier Paper 1H 82 66 47 28 14

2H Higher tier Paper 2H 82 67 48 29 14

(Marks for papers 1F, 2F, 3H and 4H are each out of 100.)

1MA0 A* A B C D E F G1MA0F Foundation tier 141 121 101 82 63

1MA0H Higher tier 164 133 95 57 28

(Marks for 1MA0F and 1MA0H are each out of 200.)

Grade boundaries are set by examiners for the whole qualification at A, C and F and the intermediate grades are calculated arithmetically. Thus, for example, the overall grade for B at Higher tier falls midway between 131 and 53 at 92. By the same token the grade boundaries on each of the higher tier papers are strictly 43.5 and 48.5 but are rounded down for the purposes of the table above.

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