Examiners’ Report

November 2017

Pearson Edexcel GCSE Mathematics (1MA1)

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Principal Examiner Feedback – Foundation Paper 1

Introduction

The paper was accessible to students who had been prepared for a Foundation tier GCSE Mathematics paper. There were some questions which were not well answered, especially towards the end of the paper, but this can be expected from the cohort sitting the November paper.

The standard of work seen was good in places but students are reminded to show full working out. At times the handwriting of students was very difficult to read; numbers should be formed clearly and when explanations are required clear sentences should be written. If handwriting is an issue for a student, centres are advised to consider special arrangements. Some marks are being lost through illegible writing.

Students are also reminded that examiners cannot make a decision about which method to mark. Whilst students may try different options, it is essential they indicate which method is their final approach. This can be easily achieved by crossing out the incorrect approach. If two methods remain with no choice indicated, both methods will be marked and the lower mark will be awarded. It is not in the student’s interest to leave more than one method visible.

It was pleasing to see that students were not over thinking questions this session and were able to work with the AO1 questions in a straightforward manner.

Report on Individual Questions

Question 1

This question was accessible to students. In part (a) some students wrote 36500 as the answer and others did not convert to just metres. With m on the answer line a final answer of 3.65 was expected.

For part (b) many students used their knowledge of 1000 g = 1 kg, but the common misconception seen was that there are just 100 grams in a kilogram.

Question 2

Whilst this tested the order of operations, far too many students gave an answer of 90 and did not follow the rules of arithmetic.

Question 3

It was pleasing to see that most students knew that y = 10.5 × 4; unfortunately a significant number could not evaluate this multiplication correctly. The need for accurate basic arithmetic on a non-calculator paper is obvious but unfortunately not displayed by students often enough. This question was only worth one mark and so students with an incorrect calculation scored no marks.

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Question 4

The question was very well answered with the vast majority of students scoring the mark. The most common answer seen was −9 + 2 and students were happy to write in the answer boxes given.

Question 5

This was a well answered question with many correct answers seen.

Question 6

Students were able to access this question with many identifying the correct methods to form an expression or write an equation for L in terms of a to gain at least some marks. Many understood the need to add all terms to form an expression but were unable to gain full marks due to poor algebraic manipulation skills. Some solutions showed incomplete simplification such as L = 5a – 1 + 4 or gave an incorrect simplification such as a5 in place of 5a or a3 when beginning to write an expression.

A large number of students were able to correctly form a simplified expression but then did not convert this to an equation. Another common error was due to incorrect arithmetic when adding 4 to –1 and stating L = 3a – 5 as a final answer.

Question 7

Part (a) of this question was well answered with the majority of students giving the coordinates in the correct order. Most used the answer line appropriately some put both numbers before the comma given, this was allowed for the one mark.

In part (b)(i) the point was usually correct and labelled. A lack of labelling was condoned if this was the only point plotted on the gird. Students should be encouraged to label answers as this helps clearly communicate their intention to the marker.

Part (b)(ii) was less well answered with many students drawing incorrect lines and then saying yes. The most successful approach seen was to set up a table and give x values including 2 and show the appropriate y values. Although all these points were not needed this approach was both popular and successful. Another successful approach seen was to plot at (0, 1) and show the gradient of 4 as 1 across and 4 up twice to arrive at the point (2, 9).

For part (c) many candidates drew a horizontal line through (0, −2), i.e. y = −2, or no line at all or multiple lines. Some also just plotted the point (−2, 0) and gave no line at all.

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Question 8

This question was accessible to all students and a good proportion of fully correct answers were seen. Many students drew the rectangle 8 by 4 without any working out being shown. Some students applied the ratio 2 : 1 incorrectly and drew a rectangle 3 by 6 as a result, scoring one mark.

Other students initially struggled to find factors of 32 but resiliently kept dividing by 2 to find them, which was pleasing to see.

Question 9

This question was not well answered; many students just worked out the correct answer which did not answer the question set. The mistake must be identified to satisfy the assessment objective on this specification. Very few students identified that you should multiply 348 by 2 rather than divide.

Centres can help students by looking at questions of this style and discussing the mistakes made in the working; they can then help them to articulate the mistake rather than finding the correct numerical answer. It is also worth pointing out to students that a one mark question should not take too long and is unlikely to require two or three calculations.

Question 10

For part (a) many candidates did realise that Jake’s scores were closer together but some struggled to explain why. A few stated the ranges, although sometimes they got them incorrect and therefore failed to gain the mark. Some students summed the scores for both Sarah and Jake and told us ‘Jake because he has the highest overall total’; this clearly gained no marks as it did not answer the question. A number of students stated that ‘Jake is more consistent because he has scores of 8 and above’; this statement was not sufficient to tell us that Jake’s scores had a smaller range and so gained no marks.

A number of students also had the idea that the range is the mean of the differences between each number, so they found the differences between each number, added them together and divided the result.

It was pleasing to see that the majority of students were able to engage with the stem and leaf diagram in part (b) and identify that the value had been read incorrectly due the key not being used. Many gave the correct answer, communicating clearly that 26 was the correct mode and some even explained what error had been made, giving examples of how the key should have been used. Others discussed the fact that 9 was the only single digit number in the diagram. Incorrect answers were usually confused or agreed that the mode was 6.

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Question 11

This question was well answered with most students able to score at least 1 mark.

In part (a) the most common error was to not realise 30 ÷ 8 gave an answer of 4 adults required rather than just 3. Most students showed some working either counting in eights or giving a division if working is seen a process mark can be awarded.

In part (b) the follow through allowed an incorrect answer in part (a) to be correctly interpreted and still gain marks. A common approach seen was to consider adding the extra children to the adult without a full allocation often showing 6 + 2 = 8 or equivalent.

Part (c) seemed to be understood but sometimes the students found it hard to articulate their answer.

Question 12

This was a well answered question. The majority of students were able to answer part (a) scoring full marks and then able to go on to answer part (b) and answer that correctly also. If a slip was seen in part (a), two marks were often awarded and the follow through applied to part (b).

Most candidates were comfortable with two way tables but a small minority did reverse the entries so careful reading of the headings is required.

If part (b) was incorrect, it was often seen as the incorrect answer. Some students wrote

unlikely when a numerical answer was required.

Question 13

This question was not well answered and many responses were blank. Surface area is an expected skill and knowledge that a cube has six faces is also required, but most students failed to realise this and so did not divide 294 by 6. Even when drawing a diagram, students often still could not see the 6 faces. Some divided by 4 or 2 or even divided by 2 and divided by 4, adding the answers together, in the possible belief that this was the same as dividing by 6.

Of those few that did divide by 6 most found the answer 49, although 48 was a popular incorrect figure. However, most did not realise this was the answer for the area of a square face and did not go on to find the square root.

Some other incorrect methods included 49 × 49 × 49 and 294 × 294.

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Question 14

Students often secured some marks for this question, but very few scored full marks. A number of approaches were seen with the most successful being those who decided to convert the fractions to equivalent fractions with a common denominator (normally 35); these candidates scored well. Some students did try to use 100 as a common denominator and this approach was usually unsuccessful.

A different approach seen was to try to draw diagrams for comparison, but these were often drawn without any consistency for the fractions and scored no marks.

The other common approach seen was to convert the fractions into decimals and this was very successful for those that were able to accurately show the division calculations. Some students had difficulty with 5÷7 and so the overall approach scored a variety of marks dependent on the arithmetic skills of the student.

One mark was awarded to those students who decided to write as 1 , but unfortunately

they usually stopped at this point. A common incorrect approach seen was to say because it

was less than 1 with no consideration of the ‘gap’ to 1.

Question 15

A good number of students were able to gain full marks for this question and if not full marks

then M1 for was often awarded. A few students tried to do 20 ÷ 9 and some others added

the ratio parts incorrectly; 22 was frequently seen but this was not helpful in finding the percentage they required.

A small number of students found the percentage of the wrong colour, generally red buttons as this was the first part of the ratio; a full method for a colour other than orange was awarded a mark, but students should be reminded to read questions carefully to maximise their potential to gain marks.

Question 16

In part (a) a good number of students were awarded the mark available for stating that both values had been rounded up. However a significant number only stated that one amount was rounded, with some not stating in what way. Another error commonly made was to state what the answer should have been and therefore not communicating anything about rounding or estimation. The correct answer was not asked for; the knowledge of estimation was being assessed.

For the second part of the question a good number of fully correct answers were seen, with clear processes and good accuracy in the calculations. Of those that did not score full marks, many lost the accuracy mark due to calculation errors when carrying out the initial multiplication but were able to gain process marks for correctly finding either 10% or 90% for their figures.

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Students who chose to use a build-up or repeated addition method rather than a grid method often lost marks due to confusion when adding their values. Arithmetic errors when subtracting £20.30 from £203 were also common.

Those students who only achieved one of the four marks available rarely showed a method for calculating 10%, with £23 rather £20.30 seen frequently. Some students also correctly calculated 10% then failed to use it to find 90% and stated the 10% discount as their final answer.

A small minority of students did not read the question carefully, finding the cost of 30 t-shirts rather than 35, candidates are reminded of the importance of careful reading and then checking they have answered the question asked.

Question 17

This question was answered well by many candidates. The most successful approach was to write a list of possibilities and then extract the number of successful combination and place this number over the total number of combinations. Students who attempted a systematic listing of outcomes scored well. Some problems arose when an unsystematic listing was attempted or only a partial list was given.

was a common incorrect answer. Some mixed up the concept of odd and even and so

was another common answer, although sometimes this answer was due to a basic adding error.

Question 18

This question was well answered with many correct methods leading to the calculation of Kelan’s share of £450 seen. Most involved summing the ratio parts, dividing into 450 and multiplying by 3. A few students made errors multiplying 45 by 3. Some were only able to

identify Regan’s share since this was of 450. The most common incorrect attempts

involved dividing 450 by the 3 shares leading to the incorrect answer of 150 or obtaining one part (45) and then dividing this by 3.

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Question 19

This question was well answered by students, with many gaining full marks. The most successful approach was to find the amounts for 8 people and add this on to the amounts for 16 people. On the whole non-calculator methods were shown clearly although the workings of a minority of candidates were somewhat messy and scattered making their methods difficult to decipher.

Common errors seen were 24 – 16 = 8 and then adding this value onto the ingredients to get 128 g, 148 g, etc, multiplying the given ingredients by 2 or 24 – 16 = 8 and then multiplying by 8.

Of the candidates who attempted the question but didn’t gain full marks, many managed 1 mark for an initial step or 2 marks for a correct method to find the amount of ingredients required for one of the items given in grams.

Question 20

This question was answered in a variety of ways. An approximation was required but a few number of students tried to work out the actual calculation. This is a non–calculator paper and as such students should realise that complex arithmetic would not be set. There were also some who said 42 = 8; however, this misconception was not seen too often.

For those who tried to approximate many rounded to 600 or used 5 thus securing a mark. Others went further and got to 600 ÷ 21, some then just writing an answer of Ami or Josh without further justification which gained two marks. The most successful correct answer was to get to 600 ÷ 21 and then to show 600 ÷ 20 = 30 or 20 × 30 = 600, thus allowing the student to justifiably select Ami or 27.1115 as the appropriate answer.

Question 21

A small number of students were able to gain two marks out of three for putting all the given numbers in standard form. Some also gained two marks for 0.0018 as their answer, not realising the need to give the answer in standard form as requested.

A few students gained a mark for rewriting one number in standard form or for showing 1.8 × 10n. A common mistake seen when attempting to write numbers in standard form was, for example, 6−2 rather than 6 × 10−2.

Question 22

In part (a) very few blank responses were seen. However, the responses were mixed and mainly scored either full marks or no marks at all. A variety of methods were seen to create equivalent fractions with common denominators but a large number of students were unable

to find the correct numerators, leading them to the incorrect answer of . Students who

created fully correct equivalent fractions were sometimes then unable to add the numerators correctly but were awarded a mark for a valid method. Other common errors included simply

adding the numerators and then adding the denominators and giving as their answer.

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In part (b) it was surprising to see –8 as a common incorrect answer. This question was not well answered and centres are advised to practice the use of negative indices.

Question 23

This was well answered by a majority of students. Of those students who didn’t score well, errors included listing the factors of 36, arithmetic errors and not using the multiplication sign in their final answer.

A significant minority found the correct prime factors but then instead of multiplying they put addition signs between the factors or just listed the prime factors, both of these approaches lost the accuracy mark. Some did not factorise fully leaving 4 or 9 as an incorrect prime factor.

Question 24

Some students tried to use algebra to solve this problem but the majority of attempts seen were based on a trial and improvement approach, normally in a non-systematic way, trying to find 3 ages that fitted the relationship requirements in the question, and summed to 77. Those that worked systematically were more successful than those that did not. Some candidates were able to find sets of numbers for example 1, 8 and 16 that fitted the relationships, but did not sum to 77, this start was given 1 mark.

Those who used algebra often failed to use a single variable to base the relationship on. Students who identified Jay as being 14 then usually correctly found the age of the other two people. A few did not express this as a ratio as was required by the question. Provided the order of the ratios was made clear, variations on the correct order of ages was allowed for full marks, as was an equivalent ratio.

Question 25

This question often allowed students to score part marks as many could identify at least one angle in the correct place. Others were then able to go on to show that the angles in the triangle ABF must be 35, 75 and 70. When given, the more straightforward reasons for angles were generally clear and correct for example, opposite angles, angles in a triangle and angles on a straight line were often correctly explained. However, reasons relating to parallel lines or opposite angles in a parallelogram were often not seen and so full marks could rarely be awarded.

A minority of students incorrectly assumed that triangles ABF or DEF were isosceles. Centres should remind students of the need to ensure they clearly label their angles either on the diagram or using 3 letter notation and also to give full reasons for their working and only give the reasons they actually use.

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Question 26

This question was not well answered. A high proportion of students did not attempt the question at all.

Many of the answers seen tried to compare only the radii whilst another common response

was to say, incorrectly, that ‘one circle out of 3 is shaded so is shaded’ with no further

explanation offered. Also comments such as ‘Daisy is correct, there are 3 sections to the logo and one is shaded’ were very common.

Few students used the area of a circle formula and of those that did, it was exceedingly rare that was cancelled in any comparison or working. Where marks were gained, it was for calculating the area of one circle, usually using the radius of 10.

Centres should discuss with students that if a question appears to require multiple calculations with on a non-calculator paper, an alternative method is likely to be available.

The thought process required for this question did appear to be beyond the majority of students entered at this level in this November session.

Question 27

For part (a), a pleasing number of students used the printed table as a basis for their calculations, clearly indicating the mid-points and showed the multiplication by the frequencies.

There were quite a few students who were able to find the mid points but weren’t sure what to do next. Some went on to add the mid points together then divide by 5. A common error seen in calculations was 500 0 = 500 thus at least the accuracy mark was lost but if this was the only error all method marks could be attained.

Students should be encouraged to go through their exam papers with questions like this and check to see if their answer is sensible, a mean outside the data range is not.

Part (b) was poorly answered. Most students commented on the accuracy of the answer and not about the appropriateness of the mean average. Many just stated should have used the mode or should have used the median without explanation and this did not score the mark.

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Question 28

This question proved challenging to many students and was not attempted by some. It was very common to see responses that confused area and perimeter, so an equation involving x and y was set equal to the perimeter, which was incorrectly stated to be 48.

A small number of students used a structured approach, either involving the identification that the two expressions in x had to be equal to 48 ÷ 3 = 16, or that the two expressions in x were equal. Those who did so normally identified that x was equal to 5, although there were algebraic errors seen in attempts to solve the equation. Far more students used trial and improvement methods, and some then realised that x = 5 gave the same answer for both expressions, and then used this to show that y = 3, the requirement of the question.

The most successful start seen was to assume y = 3 and show 48 ÷ 3 = 16, or to set the expressions of the two sides equal. However, many students did not develop the solution further.

Question 29

This question was accessible to many students who showed an understanding that the error lay in the joining of the points with straight lines. They expressed this in a variety of ways but often used words such as curve or smooth line or commented on the incorrect use of a ruler.

However, incorrect answers showed a lack of confidence with plotting quadratics. There were several who thought that the wrong points had been plotted, the graph should have passed through 0 or that it should have been a straight line. Other common wrong answers did not relate to the graph itself but on the lack of a title or a table of values.

Question 30

Most students attempted this question but many did not recognise the question as a reverse percentage and for those that did, they were often unable to find a complete method for the question. The most common incorrect approach was to increase £2.80 by 30%. The value of 4 was also seen from obviously incorrect working and so using the marking principles no marks could be awarded in this case.

Summary

Based on their performance in this paper, students should:

learn metric conversions, e.g. 1 kg = 1000 g; ensure that their basic arithmetic is sound; memorise standard formulae for this new specification; practice working through ‘explain’ and ‘give reason’ type questions; read each question carefully to ensure that the final answer does answer the set question.

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Principal Examiner Feedback – Higher Paper 1

Introduction

This paper proved to be a challenge for the cohort of students who entered. The majority of the questions in the second half of the paper yielded almost no marks for most students; not surprising given that the majority of entries will have been from students presumably aiming for a grade 4 rather than a grade 7 or higher. It is possible that a good number of the students entered would have had more successes on the Foundation tier paper. However, there were a number of questions where students were able to gain marks.

Report on Individual Questions

Question 1

A familiar question to students, and one where a larger percentage of students were able to score full marks for a correct prime factor decomposition, either using index form or not. When full marks were not scored many gained one mark for a correct partial method.

Question 2

The most successful approach to this first problem question on the paper was to form an equation based on the ages of three friends, solve to find the ages, and then write the ages as a ratio. Most students attempted an algebraic approach, but often only managing a single mark for two correct expressions. Once expressions had been formed some students were then able to sum and equate to 77 to gain the second mark. Those who managed to form an equation, were normally able to gain the third mark for isolating their terms, or more commonly for actually solving to get 14. Of those who got this far many found all three ages and formed a suitable ratio. However, there were quite a number who forgot the demand of the question and lost a mark as they did not present their final answer as a ratio.

Question 3

Students continue to struggle with geometry questions in forming complete solutions with full reasoning. There were a great number of methods evident within the responses seen, and most students were able to score some marks. For the B1 it was typically for 35 marked on the diagram at AFB, although any correct angle marked gained this mark. 2 marks was a common score, either B1C1 for an angle marked and a correct reason, or for a correct pair marked as per the mark scheme. However, many who scored B1M1 also scored C1 for an appropriate reason. It continues to be disappointing to see students with fully correct working drop marks due to incorrect reasons stated, or reasons incorrectly stated.

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Question 4

The second problem question on the paper proved to be a huge challenge for many in this cohort. In particular students providing just a written explanation relating to one circle out of

three, so it is . Those who started to work with π r 2 typically got at least two marks for a

method to find the shaded area, which was very pleasing to see. Of those who got this far, it

was about even as to whether they could then draw a correct conclusion and compare to .

Noticeable was the error that occurred when students believed to be equal to .

Question 5

It was surprising to see part (a) of this question, a common one on the legacy specification, answered so poorly. Many students appeared to not understand estimating the mean at all and made no attempt to find fx. When students did it was most common to see two marks awarded. Some dropped the second by failing to divide by 20, many making the common error and dividing by 5. Many who scored two marks didn’t gain the third mark due to arithmetic errors, typically when finding their products rather than the division.

Part (b) of this question proved to be too much for almost all students. It seems that more time needs to be taken within centres on choosing the most suitable average. Very few students gained the mark on this question in which they had to consider the effect of outliers on the mean, typically just stating that the mode or median would be better with no suitable reason why.

Question 6

This problem put a slightly different twist on a familiar concept, but was one that most students were able to access in one way or another. The most common approach was to equate the two lengths and solve for x, before substituting into one length and using the area to find y. Of those who were unable to form a suitable equation, many gained a mark for finding the width by assuming y = 3, for example 48 ÷ 3 = 16. There were a number of students who scored zero as they went down a route of finding the product of the two lengths rather than equating them, or worked with perimeter.

Question 7

Students responded well to this question and a large number realised that the use of line segments on a quadratic graph was wrong. There were, though, many students who talked about mis-plotting of points, or who wrongly thought that the graph should go through the origin.

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Question 8

A question ordering recurring decimals has never been set before, but most students were able to gain some credit. One mark was the most common score seen; this was normally gained for having three of the decimals in the correct relative order, although many gained the mark for showing understanding of the notation, for example 0.2464646…

Question 9

This speed problem was too much for many students, and typically the best score seen was one mark only. This was awarded for a method to find the correct speed for James, for example 50 ÷ 2.5 or, in minutes, 50 ÷ 150. Unfortunately many of those who attempted this were unable to divide by 2.5, or when working in minutes completed the division the wrong way round and were unable to get further. A small number of students were able to get further, using time in hours and minutes to get to Peter’s 40 minutes for 15 km. Those who did get this far generally went on to get the correct answer.

Question 10

Part (a) of this question was answered quite well, with a good number knowing that the power

of is the square root. The common incorrect response was to halve 100 and get 50.

In part (b) fractional indices to this degree proved harder. There were a large number of

students who dealt with the power simply as a fraction and attempted to find of 125.

However, getting as far as 5 allowed many to score one mark. A small proportion of students were able to complete the solution to 25.

Question 11

Forming and solving simultaneous equations proved to be where many students stopped gaining marks. Many students attempted to solve this problem through a trial and improvement method, normally with little or no success. Of those who gained a mark for forming two equations, many then had no strategy for solving them. Those who did have a strategy often made arithmetic errors leading to incorrect answers.

Question 12

Most students gained at least one mark for three correct values on the box plot in part (a). Those who had a good understanding gained three marks, and this was commonplace. There were a number of students who plotted an incorrect value for their minimum; this normally meant no marks were scored as all the values were then in the wrong place on the plot.

In part (b) students struggled to gain marks for comparisons of plots. Some students just stated values but made no comparisons. Some stated figures (which they didn’t need to do) and stated them incorrectly within a comparison and thus lost marks. Typically, those who made comparisons gained only one mark as they were unable to contextualise their statements.

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Question 13

The concepts of proportion within part (a) of this question were too complicated for most

students; many started with the fraction of rather than , and as a result failed to score

marks. Those who started correctly normally scored full marks.

Part (b) was very poorly answered with almost no one scoring two marks. For those who scored 1 mark, it was normally for stating a correct bound (5.5 or 6.5).

Question 14

Students typically tried to start with the solution here and worked backwards, consequently

scoring no marks at all. Very few students started with a correct statement, for example  

= k. Those who did very often had sufficient skill in manipulation to gain full marks.

Question 15

Another familiar question from the legacy specification and one where we would expect to see more correct algebra. Most students understood the need to find multiples of x, unfortunately these were either often wrong, or the wrong multiples were found; for example, finding 1000x but not 10x. There were, though, a good number of students that were able to follow the algebra through to the correct fraction.

Question 16

Most students did not know how to structure their response to this fairly standard question, with a significant number being unable to form a suitable equation. Those who did were then

troubled significantly by the fraction and struggled to be able to divide this by . It was

pleasing, however, to see students showing their working and as such, those who showed they needed to complete this division were able to score two out of three marks.

Question 17

Most students understood the need to expand the brackets, but with only two marks available, both had to be expanded correctly to gain the method mark. This proved a step too far for many with one or both being expanded incorrectly. Those who did expand correctly were very rarely able to simplify and factorise correctly to complete the proof.

Question 18

A very disappointing performance on this enlargement question. Most students completed

either an enlargement of scale factor 2 or , and as such scored no marks. Few had a method

or understanding to arrive at a triangle in the correct orientation and size which resulted in very few marks being awarded.

Question 19

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Almost no students were able to start this question at all. The few that gained any marks typically rearranged the given equation into the form y = mx + c to find its gradient, or for showing an understanding perpendicular gradients. The concepts appeared to be beyond the vast majority of the cohort.

Question 20

A slightly better performance was seen here and it was evident that some effort has been made in centres to get students to learn the values of the trigonometric ratios, or how to find them using unit triangles. This meant that a reasonable number gained one mark for a correct value of cos x. Very few were able to use this knowledge, combined with the information in the table, to form any equations in a and b. As such more than one mark was rarely awarded.

Question 21

Again, very few students understood how to rationalise a complicated denominator such as this. Those who had some understanding often used 2 – 1 rather than 2 + 1 and therefore gained no credit. Those who knew which surd expression to use were normally able to gain at least two marks for the correct expansion of the numerator, many being unable to then simplify fully.

Question 22

It was good to see a number of students having some understanding of similar shapes and gaining two marks for a value of x = 2 correctly found. Almost no students were then able to deduce the second assumption that could be made and as such no further marks were awarded.

Question 23

This question did allow for some marks to be awarded to a small number of students who were able to write expressions for both areas. However, it was disappointing to see how many failed to gain this first mark because they failed to use the correct formula for the area of a triangle, often forgetting the half. Very few were able to form a suitable inequality from their expressions, of those who did and had strategies to solve (typically by factorisation) almost no

one gained full marks as they failed to discard the critical value of .

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Summary

Based on their performance on this paper, students should:

focus attention on basic arithmetic skills such as multiplication and division, and multiplying by powers of 10 to ensure marks are not dropped unnecessarily;

practise non-standard procedure (AO1) questions and be able to answer them confidently – for example, finding the mean from a grouped frequency table.

spend more time working with ratio and proportion;

practise solving problems where translating the given information into algebraic expressions and equations is an efficient method of solution.

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Principal Examiner Feedback – Foundation Paper 2

Introduction

A significant minority of students found this paper difficult, and were clearly unprepared for some of the questions. Indeed, it was clear that sometimes it was the content rather than the application of mathematics that was a problem for the students. Performance was not always consistently good across the paper, but with a good range of questions the paper was able to discriminate well.

Weakest areas included algebraic manipulation and derivation, percentage calculation and application of ratios and rates. There was also evidence that a number of students did not have a ruler for measuring lines, or a protractor for measuring angles. Most demonstrated the use of a calculator, though on some occasions it was clear that they did not have an understanding of the way in which their calculator worked. Unfortunately, some answers were spoilt by premature rounding, or taking an accurate answer from a calculator and rounding it sufficiently to make it inaccurate.

Questions which assessed the use of mathematics across a range of aspects of the specification were sometimes done poorly, such as 15, 18 and 22, but in other cases done well, such as in question 11. There was also inconsistency of approach to questions that might be considered more traditional where the process of solution might be considered predictable, such as poor attempts in questions 4, 5 and 16, yet good attempts at 13 and 17.

There were far fewer attempts using trial and improvement approaches. Approaches to questions that required some interpretation or explanation were inconsistent. Questions 4(b) and 8 were questions in which most students scored well, but poor attempts were made in 14 and 18.

Students need to read the questions carefully. There were too many case where students misread the question and failed to give the answer asked for; equally too many cases where figures given in the question (and sometimes in their own working) were misread.

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. This is most important in longer questions, and in “show that” questions. Examiners reported frequent difficulty in interpreting complex responses, poorly laid out, in quesitons 15, 17, 19 and 22.

Report on individual questions

Question 1

A well answered question to start, where the only common error was a misplaced decimal point, for example 0.7 or 0.007.

Question 2

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Nearly all students gained this mark.

Question 3

In part (a) only a minority failed to simplify and left a “×” sign in.

In part (b) weaker students wrote their answer as t2 rather than t 2.

In part (c) it was encouraging to find most students were able to simplify the expression.

Question 4

Part (a) was generally well done, with the common error in not carrying out the final division by 90 correctly. Some students failed to extract all the information correctly, for example not finding the total weight of each item of fruit.

In part (b) the most popular choice of method was 75 × 15 with a conclusion. Some students showed 13 tomatoes at 975 and 14 tomatoes at 1050. The conclusion was sometimes written in part (ii), but marks were still credited for this in part (i).

The last part was very poorly done, with many students just repeating what they had done in part (i); it was clear few understand the mathematical concept of an assumption.

Question 5

Students had a good success rate with part (a), most giving as their answer, but some did

simplify. The most common error was giving the probability of those that did walk. Some failed to read the 60 in the question and added the total incorrectly.

In part (b) it was disappointing to see so little working shown, which would have gained many students an additional mark. There was evidence that students did not have a protractor, particularly when they showed the angles, but were unable to draw any angles accurately. Most students did show a pie chart with at least two angles drawn accurately.

Question 6

In both parts of this question there was confusion as to which numbers to include in their

answer. In part (a) answers ranged from , and the incorrect fraction for Annie.

In part (b) there was similar confusion with ratio being given the wrong way around (for example 1 : 3) or use of the wrong numbers (for example 1 : 4, 3 : 4).

Question 7

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A large proportion of students gained full marks by correctly identifying the relevant prime numbers, but some failed in this process by including a non-prime number. Weaker students confused prime numbers with square numbers, and sometimes included numbers between 1 and 10.

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Question 8

In part (a) a minority of students recognised the missing “30” on the vertical axis; too many made reference to the diagram as if it were a scatter graph. But in part (b) it was the majority who correctly identified the trend as increasing, though some answers were spoilt by incorrectly referring to the trend as “positive correlation”.

Question 9

Part (a) was done well, with the most common answer being 2.75. Most showed 5.5 × 0.5; there was some evidence that students did not have a ruler, and were guessing the length. There were a few students who, having shown 5.5 × 0.5, then gave their answer as 3. Students should be encouraged to write their accurate answer and not round it.

In part (b) some students appeared confused as to which angle they were giving the bearing for. But even when it was clear the correct angle was being found, the protractor was being read wrong (for example 130). There was also some evidence that students were estimating the angle (perhaps because they did not have a protractor).

Question 10

In part (a) is was not uncommon to find students missing off the units from their answer; this was far more frequent than those giving an incorrect unit (such as cm). There was some evidence of counting squares to get to the numerical answer, but this usually led to an inaccurate value. It was disappointing that a significant minority did not halve their value of 24.

In part (b) many students had difficulty in naming the shape, giving almost any quadrilateral other than “kite”.

Question 11

Most students were able to arrive at the correct values of 250, 100 and 500 to put into a ratio, though many then failed to simplify it correctly or fully. Some errors were seen in arriving at these values, most often the “100”.

Question 12

This was very well understood and full marks were gained by the vast majority of students in both parts.

In part (a) a minority of students put the frequencies in the wrong order in (usually) two of the right-hand boxes; some gave the frequencies incorrectly as probabilities of 200.

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In part (b) the most common incorrect answer was (which gained one mark) where

students had not read the question properly. Very rarely did students use incorrect notation for the probability.

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Question 13

This question was well answered. Students calculated the hourly rate for both people or calculated the equivalent amount earned for comparison with the £266; some did both! It was disappointing to find a common misread of the £266, usually replaced with £226.

Question 14

In part (a) students had to give an example by choosing two odd numbers in the given expression, and calculate an answer that was a multiple of 4. This was usually done well, sometimes using the same odd number, though errors were not uncommon. It was surprising to see even, rather than odd numbers being used, and merely substituted and not worked out.

In part (b) students had to show some reasoning by explaining how the use of (any) odd number in the expression could give a multiple of 4. Marks in this part were rare. Nearly all students thought that they just had to give multiple examples as in part (a). Some gained the first mark (only) by reasoning that doubling an odd number always gave an even number.

Question 15

Students had to read and analyse the information given in this question, and then formulate a strategy for its solution; for many this caused too many problems. The most common misconception was with students finding an incorrect amount of oil purchased in both November and February with very few finding the 1500 and then subsequently adding 400 to this value to gain the total value of 1900. Many gained the process mark for showing a method to find an increase of 4%, though again there were many common errors including use of a 1.4 multiplier, division by 4 and partitioning methods that failed to add to give a 4% increase. Very few students provided a full solution leading to the correct answer.

Question 16

This question differentiated well across the ability range. Both trial and improvement and the flow chart method rarely resulted in the correct answer. Many gained a mark for expanding the bracket, but most then were unable to perform a correct manipulation of terms to get the

second mark. Some students stated the answer as rather than .

Question 17

Responses to this question started well. Most were able to calculate the profit on either one bottle or 12 bottles. The £0.36 profit was often seen. A significant number of students stopped there, sometimes giving 0.36 as their answer. Many students erroneously took the base for comparison as their selling price, £6 rather than £5.64 cost price. Some appeared to get as far as 1.063 but then rounded to 1.1.

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Question 18

Few students scored any marks in part (a), with the commonest answer being 10, from .

Some scored one mark for working with the circumference; no marks were gained from using the area formula.

Part (b) was better answered with about half gaining credit from recognising that some aspects remained constant. Unfortunately, many answered “yes” because they thought the circumference had changed.

Question 19

When students worked with numbers of cubes initially they were more successful. Some students began by incorrectly writing the ratio of Y:B the wrong way around; another common mistake was to interpret the green cubes as 4 × yellow (instead of 4 × blue). Some algebraic attempts were seen but these were rare and lead to many mistakes in calculating G. The most successful methods tended to use possible numbers of each colour, for example 4, 2, 16.

It was not uncommon for students to use their own chosen values to represent the ratios which could also lead to the correct answer. Some marks were lost when working was not clear, or was sometimes contradictory when multiple methods were presented.

Question 20

Many students met with some success in this question. In part (a) there were some students who rotated the shape by 90° rather than 180°. Sometimes the shape was not accurately drawn in the correct position.

In part (b) students were not careful enough counting squares, and sometimes positioned the shape within one square of what was needed. Some failed to take account of the minus signs in determining direction of move.

Question 21

In part (a) addition was required; some multiplied and gave the answer as 3.

In part (b) multiplication was required; some added and gave the answer as 8.

Students very rarely scored marks in part (c). A clear lack of conceptual understanding of standard form (or indices) was evident with most students missing the powers of 10 link to gain the correct algebraic power. Many tried to solve the equation to gain a numerical value with the use of the 3 and 2 as coefficients of 100 and 1000 rather than showing their derivation.

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Question 22

Only the best students were able to work through to a correct solution, but part marks were awarded to those who attempted to do something of worth with the diagram. Some started with Pythagoras on the left hand right-angled triangle, but of course only earned marks if it was of the form 7.52 − 62 (i.e. not added). It was not uncommon to find some attempting to find the area of the trapezium, which of course earned no marks. There was some (independent) credit for working with trigonometry. This could be done in the left-hand triangle (if the angle was made clear) or in the right-hand triangle (with their stated value for the base). However, only a minority of students realised that trigonometry was needed.

Question 23

There was a poor success rate for part (a). It was clear that some were just entering figures on their calculator without any forethought as to how to get the calculator to process part values. Those who worked out the four values and wrote them out, then moving on to the rest of the process of calculation frequently gained the final correct answer. Some lost marks due to premature rounding of the figures from their calculator.

Good rounding in part (b) frequently led to the mark in this part being awarded. There were errors for some who used the wrong number of decimal places.

Question 24

In part (a) sight of a complete answer (both 6 and −6) was rare. Some credit was given where an answer was embedded, which was not uncommon. The main mistake occurred when students divided by 2 twice instead of dividing by 2 and then finding the square root.

In part (b) the majority were familiar with what was required but many failed to multiply the 3x by 3x correctly, often writing this as 6x, but gained one mark if they multiplied their other terms correctly. Using a table format was very popular and generally successful for those students.

In part (c) very few were familiar with the requirements of factorising into two brackets so often tried to “factorise” using only one pair of brackets. Common wrong answers like x(x + 6) + 9 were frequently seen.

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Summary

Based on their performance in this paper, students should:

• present their working legibly and in an organised way on the page, sufficient that the or-der of the process of solution is clear and unambiguous;

• include working out to support their answers;

• bring all necessary equipment to the examination, including ruler and protractor, and be trained in the correct use of their calculator;

• carry out a common sense check on the answers to calculations; for example they should have expected the number of £1 coins in question 6 to be a whole number;

• make sure they learn and understand algebraic manipulation and derivation, percentage calculations and application of ratios and rates when preparing for future examinations;

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• spend more time ensuring they read the fine detail of the question to avoid giving answers that do not answer the question, and to ensure they use the correct figures as given in the question.

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Principal Examiner Feedback – Higher Paper 2

Introduction

The time allowed for the examination appears to have been sufficient for students to complete this paper and most students seemed to have access to the equipment needed for the exam.

The paper gave the opportunity for students of all abilities suited to entry at higher tier to demonstrate positive achievement. Though it was to be expected that most of the students presented for this resit opportunity were of modest ability, it is disappointing to report that there seemed to be a significant number of students who scored very few marks. They may have been more appropriately entered for the foundation tier paper. It was rare for students to make successful attempts at most of the questions on the paper. Few students were able to work confidently on questions 18 to 23.

Many students set out their working in a clear, logical manner. It is encouraging to report that students who did not give fully correct answers often obtained marks for showing a correct process or method.

Report on individual questions

Question 1

There were many fully correct solutions seen. Where full marks were not achieved, one mark was often awarded for a correct expansion of the brackets. Unfortunately this was frequently followed by an incorrect attempt to isolate ether the terms in x or the constant terms, the constant terms presenting the greater difficulty. Some students wrongly expanded the brackets as 3x − 1 but were then able to get some credit for the next stage in the manipulation of their equation.

Question 2

Students almost always earned some credit for a correct start to the processes needed by either finding how much Emily paid for one bottle of water or how much she got for selling all twelve bottles of water. Finding the percentage profit proved a greater challenge to most

students, a common error being to use or instead of or . A number of

the students who did use a correct fraction and got 1.0638 or 106.38 were unable to interpret this as a 6.38% percentage profit.

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Question 3

For those students who realised that they needed to use the formula for the circumference of a circle, and could recall it, this question proved to be straightforward. It was surprising to see that a significant number of students instead opted to divide the diameter by 8 to give 10 as their answer.

In part (b) students who stated that the total distance or that the number of points remained constant were awarded the mark available. It was encouraging to see that this part of the question was answered quite well, sometimes even after an incorrect response to part (a). Students generally gave a clear decision coupled with a clearly expressed reason. Where this was not the case, students often referred to larger and smaller gaps between the points. Students who thought that the mean distance would change used the reason that the points would not now be equally spaced.

Question 4

This question discriminated well between students of different abilities. The weakest students merely rephrased the question, for example by writing B = 2Y. Students going no further than this could not be awarded any marks. Most students did gain some credit for their attempts for writing down a ratio linking the number of cubes for at least two of the three colours. More able students wrote down a correct ratio linking all three colours. A common error was the use

of the incorrect ratio Y : B : G = 1 : 2 : 4 leading to the incorrect answer, . An alternative

strategy used by many students was to assign particular numbers of cubes to satisfy the

conditions given and work from these. was a commonly seen acceptable answer as an

alternative for the correct probability in its simplest form, . The general marking guidance

given in the mark scheme states that probabilities given correct to at least 2 decimal places are also acceptable, so 0.09 was also accepted as a final answer.

Question 5

Students found part (a) involving the rotation of the trapezium more accessible than part (b) where they had to translate the shape. However, it was disappointing to see the number of responses to part (a) where the trapezium A was placed in an incorrect quadrant and examiners were left wondering why more students had not used tracing paper to help them.

Part (b) was not well done.

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Question 6

Most students were successful in parts (a) and (b) of this question. In part (a) the most common incorrect response was “3”, with “8” occurring quite frequently as a response in part (b).

Part (c) was not well answered. Only a small proportion of students stated that 100a can be written as 102a or that 1000b can be written as 103b or that 100 = 102 and 1000 = 103 and scored the mark for a first stage in the reasoning needed. Few students were able to complete this part of the question by showing that the product 102a 103b leads to the given result.

Question 7

A good number of students did realise that this problem required the use of Pythagoras’ theorem and trigonometry but were not always able to apply them correctly. Students who realised that using Pythagoras’ theorem was the most efficient way to start the problem usually gained the first two marks, although some students calculated 7.5² + 6² instead of 7.5² − 6². A common error was for students to subtract 10 from 24 to find the missing base length of the right-angled triangle needed to find angle CDA. Students who clearly showed a correct use of trigonometry to find angle CDA were awarded a process mark even if they had used an incorrect value for this length, for example 14. It was encouraging to see a good number of fully correct solutions to this multi-step question.

Question 8

A large proportion of students were successful with this question. However, many students were unable to apply the correct order of operations and did not apply the square root to the full numerical expression or were unable to use their calculator to get a correct value for the expression within the square root. Students are advised to give themselves more practice using the bracket function on their calculator. Some students omitted to calculate the square root altogether and simply gave their final answer as 7.5958…

Part (b) of the question was done surprisingly badly. A significant number of students simply truncated their answer to part (a) to 2 decimal places, and often those students applying rules for rounding not only increased the second decimal place by one but also their first decimal place by one. A significant number of those students rounding 7.597… gave their final answer as 7.6, instead of the correct 7.60.

Question 9

Fully successful solutions to this question involving inverse proportion were not common. Most of the students who did score full marks found the total number of hours needed for the 5 cleaners to clean all the rooms in the hotel (5 × 4.5), then divided by 3 to find the number of hours needed by each of the 3 cleaners. Of those students who successfully calculated the 7.5 hours, most of them correctly rounded this to 8 before calculating what each cleaner was

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paid. However, a significant minority of students calculated 7.5 × £8.20 and scored two out of three marks.

Weaker students often used direct proportion and did not question why 3 cleaners would take less time than 5 cleaners to clean all the rooms in the hotel.Question 10

This question discriminated well between students. Many students could state the time interval when the speed was greatest and a good proportion of those students were able to explain why, usually referring to the gradient of the lines. However a significant number of students misunderstood what the question required in part (a). The question asked for “two times” between which the speed was greatest. Some students interpreted this as a request for two answers and gave the two time intervals 0 – 20 and 20 – 60.

Students who scored the marks in part (a) were often successful with part (b) of the question. However, answers to part (b) were often marred by errors made in reading accurately from the graph with many students using “380” instead of “360”.

Question 11

There were very few fully correct answers to the problem posed by this question. A significant minority of students realized that the sector angle was needed and some students started to work with the areas of the circles. One mark was awarded for this.

Question 12

Many students were able to score one mark for calculating the size of an exterior angle of the regular polygon (360 ÷ 12) or for calculating the size of an interior angle (1800 ÷ 12) and about a half of these students were able to complete the question correctly to find the required angle. Evidence of confusion between exterior and interior angles was relatively rare. A significant number of students gave their final answer as 150, suggesting, possibly, that they didn’t understand the angle notation used.

Question 13

Part (a) was a straight forward application of compound growth. It was well answered by a good proportion of students but a surprising number of students did the calculation long-hand by calculating eight separate increases of 2%. Of those students who used the more concise method of using a multiplier, some used 1.2 instead of 1.02.

A significant number of students did not give their final answer correct to the nearest £100. The most common error in this part of the question was to treat the problem as a simple interest calculation.

Part (b) of the question was found to be more demanding and only a small minority of students were able to set up the problem correctly, for example by writing down an equation

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such as 250 000 × y6 = 325 000. Rather than dividing 325 000 by 250 000 to find a multiplier for the 6 year period, students often starting by working out the difference (75 000) in these amounts and then could make no further progress. A significant number of those students who did follow a correct method to find the correct multiplier for one year (1.045) did not then go on to interpret this in terms of a yearly percentage increase (4.5%). Trial and error approaches were sometimes successful but often led to poor accuracy in the final answer.

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Question 14

This question discriminated well between the more able students. Many of these students scored one or two marks for drawing 2 or 3 lines correctly. Where two of the three lines were drawn correctly it was often y = 2x which was incorrect or not attempted. Students who drew all 3 lines correctly more often than not opted for the closed region bounded by these lines rather than the open region satisfying the three inequalities given. Weaker students could often only draw the line y = 1 correctly. Some students did not attempt to answer the question.

Question 15

For part (a) of this question examiners expected students to make a decision about whether Tracey is correct and then explain that the numbers 8 and 7 needed to be multiplied, and not added, to work out the different number of ways of choosing a main course and a dessert. The question was quite well answered with many students giving clear and concise answers though some responses were too vague to be awarded the mark. For example, some students merely stated that there would be more than 15 ways with no further explanation.

Many different approaches to work out the total number of games played were seen in part (b) of the question. Some students used a listing method or a diagram, equivalent to adding the integers from 1 to 11 inclusive. Students who used a multiplication method often calculated 12 × 11 and did not take into consideration that this would include each team playing each other team twice; “132” was consequently a commonly seen incorrect answer. Examiners were able to give some credit for this answer. Other incorrect responses seen included 144 (12 × 12), 72 ((1212)2), 78 (12 + 11 + …+ 1) and 24 (12 + 12).

Question 16

Few students obtained full marks. The direct approach of taking square roots of each side of the equation was rarely seen. A more common approach was for students to expand and simplify (x − 2)2, then use the quadratic formula. Some students who expanded (x − 2)2

seemed to run out of steam and did not attempt to solve their resulting equation. Another very common error seen was for students to expand (x − 2)2 as x2 − 4 and find a value or values of x from the resulting two term quadratic equation. Some students who worked accurately gave only one correct value for x (usually 3.73).

Question 17

The majority of students drew frequency diagrams for part (a) this question, usually with bars of the correct width. Of those students who correctly calculated and used frequency density, some did not label the vertical axes correctly. Class intervals were sometimes used on the horizontal axis (rather than a linear scale).

In part (b) some students were able to calculate 123 by using the table even when they had drawn a frequency diagram in the first part of the question. A significant number of students, having calculated 123 correctly, did not then go on to express this as a fraction of 150. Some

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students who had drawn a correct histogram attempted to calculate the probability in part (b) from frequency densities.

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Question 18

The value of k required in this question involving an iterative process was 0.98; “98%” was not an acceptable answer. Some students did more than was expected and used the iterative process to calculate the value of V1.

Question 19

There were a small number of excellent proofs seen usually using gradients to show that the lines were parallel. Students who attempted the question but could not provide a full solution often gained one mark for a correct method to calculate the coordinates of at least one of the points M or N. Students often drew a diagram but without further work, these could rarely be awarded any marks.

Question 20

Some students were able to score one mark for calculating the area of the sector or for identifying a right angle between a radius and a tangent or two marks for both. A significant number of students wrote down a correct expression for the area of a circle of radius 10 cm but then did not work out the correct fraction of the circle. Few students were able to give a correct method to find a length in order to calculate the area of the kite. There were a relatively small number of fully correct answers.

Question 21

Only a small minority of students calculated the correct probability in part (a). In fact, not many students were even able get as far as multiplying three probabilities together and those that did often calculated or equivalent. Some students attempted to use tree diagrams but these were usually incomplete or incorrect.

In part (b) a few students were able to set up a hypothetical number of counters in the bag, usually 5 red, 5 blue and 5 yellow counters and then calculate a probability for comparison. Most of these students clearly stated their decision based on a correct comparison of probabilities. However, when comparing fractions, some students did not write them in a suitable form by using the same numerators or the same denominators or by converting the fractions to decimals.

Question 22

Though there were some fully correct answers, these were rarely seen. Few students were able to set up the required simultaneous equations, though some were able to score marks for

3a + b = 20, g(1) = a + b or for f–1(33) = 6. Some students confused f–1(x) with .

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Question 23

The vast majority of students could make no progress with this question designed to test top grade students. Some students confused the geometric sequence with an arithmetic sequence and involved addition of the terms (rather than multiplication).

For part (b) there were again few attempts worth any credit with some students starting their working by using their calculator to write down the value of 7 + 52 as a decimal.

The best students gave clear, concise and full solutions to this question.

Summary

Based on their performance in this paper, students should:

• practise solving linear equations;

• learn standard techniques such as working out values in problems on compound growth by using a multiplier method;

• carry out a common sense check on the answers to calculations, for example, expect the number of hours each of 3 cleaners need to clean all the rooms in a hotel to be more than the number of hours that each of 5 cleaners need to clean the same number of rooms;

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• use tracing paper to help in questions involving rotations;

• check any readings taken from graphs to make sure scales on the axes have been inter-preted correctly.

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Principal Examiner Feedback – Foundation Paper 3

Introduction

The vast majority of students seemed to be entered at the appropriate level and coverage of the specification was good. Students were generally well prepared, however many clearly had little idea of the nature of an error interval in 23(b). It was also very clear that there were several other questions on the paper that were beyond the capability of many students at this level, namely questions 22, 24 and 26(b).

Very few students showed evidence of not having access to a calculator. There was, however, evidence of some students not having a ruler. This was needed for drawing straight lines in two graphical questions, 13 and 19(b), as well as for measuring in question 8.

Students do not always appear to know when to show calculations and a conclusion and when to write a statement for their answer. Presentation of work was on the whole very good and few scripts proved difficult to read.

This paper did identify a concern with many candidates not knowing standard conversions, for example pence to pounds, millilitres to litres, centimetres to metres, grams to kilograms and minutes to hours.

Report on individual questions

Question 1

Very few incorrect answers here although 3800 was a common error.

Question 2

Most students were able to give an acceptable form of the correct answer. However, some did not fully simplify their answer; for example, a few students gave an answer of 4y – 2y or y + y and gained no credit.

Question 3

The majority of students scored at least one mark in this question for identifying at least 3 factors of 18. A significant number of students failed to include 1 and/or 18 in their list. Pleasingly, unlike in previous years, candidates clearly showed that they knew the difference between a factor and a multiple.

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Question 4

This was answered well and many fully correct solutions were seen. On many occasions, incorrect monetary notation prevented full marks; it was common to see answers of 0.90p or just 0.90. £1.10 was a common incorrect answer found as a result of incorrect subtraction.

Many students failed to read the demand of the question carefully enough and after fully correct arithmetic, often simply said “the family ticket is the cheaper”, without working out the required difference in cost. A significant number of students never actually showed their final subtraction and so incorrect differences quoted failed to gain any the second method marks since the method was never seen.

Some candidates worked out the cost of 4 child tickets and one adult ticket or two child tickets and one adult ticket. These were deemed inappropriate attempts and received no credit. Despite this being a paper where a calculator was expected to be used, there were errors in calculating 5.80 + 5.80 + 5.80 + 7.80.

Part (b) was also answered well. Students who realised that 102 minutes is equal to 1 hour 42 minutes usually went on to correctly complete the solution. Some converted 102 minutes to 1 hour and 2 minutes and failed to score, and some students converted to a decimal, 1.7, gaining some credit but rarely full marks.

There was also evidence of poor mental arithmetic, with addition errors common. This should not happen on a calculator paper when students can, and should be encouraged to, check their calculations.

Question 5

A significant number of students converted 2 litres incorrectly to 200ml. Such attempts failed to gain any credit since this error trivialised the question. Others realised that division was required, but did 150 divided by 2 to reach 75. However, many students demonstrated a good understanding of what was required and either divided 2000 by 150 or used build up methods to reach the 2000 ml. Answers then varied, many giving 14 (often from premature rounding of

to 7) and sometimes 12 as their answer and others gave an answer of 13.3 not realising

that they needed a whole number answer.

Many responses attempted a method of calculating 1 litre; some did not complete the process of doubling up (giving an answer of 6) or failed to account for the extra bottle when they did double their answer.

Question 6

The vast majority of students gained at least two marks and usually three in this question. It was common to see the three quarters (on Saturday) split into one half and one quarter circles. This was then often shown in the key. This was perfectly acceptable.

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Some students defined their own, incorrect key and changed the whole diagram accordingly. This received no credit. Centres should note that a key in words without showing an appropriate diagram, was not acceptable.Question 7

Although a few students tried to show that the triangle was isosceles by assuming that it was and working ‘backwards’, most students did try to find the size of the required angles BCA and CAB, using a standard approach. Quite often reasons were omitted, or those given were incomplete, usually omitting the word ‘angles’.

Many students tried to use ‘angles on a straight line’ to find angle BCD, only to show incorrect working of 180 – 117 – 54 = 9 Subsequent working to find angle CAB was then incorrect and gained no credit.

Having correctly found the size of the required angles, together with correct reasoning, many failed to complete the solution with a statement explaining why the triangle is isosceles. There were regular mentions of two equal sides but not angles. Many candidates just explained that the triangle is isosceles because it has two equal angles without also giving the reasons relating to how they calculated the angles. Others gave the reason that an isosceles triangle has two equal sides rather than two equal angles. Some students incorrectly assumed that angle BCA was 54 stating the reason ‘base angles of an isosceles are equal’.

Question 8

30 m was the modal correct answer for this question, where the height of the building was estimated at two and a half bus lengths.

Many students however used the height of the bus for comparison with the height of the building and made many errors. It was not enough for the award of the method mark to simply say that there were about 6 to 7 bus heights equivalent to the height of the building. Students had first to make a sensible comparison between the length of the bus and its height, possibly using the scale of 2 cm = 12 m.

Question 9

At least one mark was gained by most students for either correctly identifying two prime numbers or two numbers whose sum is a square number less than 30.

An answer of 1 and 3 was a common error where students assumed the number 1 to be a prime number, similarly 7 and 9 was quite common with many students regarding 9 as a prime number.

Question 10

This question was poorly answered with many students simply finding either five sixths of 48 (= 40) or two thirds of 48 (= 32). A few attempted to use decimals usually getting inaccurate answers due to premature rounding.

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Question 11

Most students gained one mark for a correct cost of £264 for Offer 1. Those failing to get this answer usually ignored the free lesson and just worked out £24 × 12 (£288).

Working out the cost of 12 lessons with the 5% discount in Offer 2 proved more of a challenge. Some used 50% instead of 5%, a common error seen was simply subtracting 0.05 or 0.95 from 24 or from 288 and some calculated the cost of 11 lessons with this offer.

Others thought “off” means “of” and assumed the cost of a driving lesson in Offer 2 was 5% of £24. A few students having shown fully correct calculations, failed to answer the question and quote the cheaper offer thus denying themselves an easy final mark. A small number did not state ‘Offer 1’ and just circled their choice. Students must understand that this is not acceptable.

Question 12

Attempts to find the cost of 1 kg (or 0.5 kg) of apples were usually accurate and usually lead to the correct answer. The most common error was to find one and a half lots of £3.60 and £5.40 was the modal incorrect answer. Some candidates simply added £1 to £3.60 assuming that the difference was the ‘same’ as that between 2.5 kg and 3.5 kg. A few candidates tried to find the weight of apples that could be bought for £1. This approach only gained any credit when a complete method was used but this was very rare.

Question 13

Accurate completion of the table of values in part (a) was common; the most popular error was in working out the value of y when x = 1, often +0.5 was seen. Surprisingly a correct graph, in part (b), often did not follow a correct table. Many students just plotted their points thinking that this was all that was required. Some started again and failed to produce a correct graph. Students should realise that a straight line graph is required and that any variation is an indication of an error. A number of students reversed the x and y coordinates when plotting.

In part (c), some students correctly found the value of x by simply solving the equation. This was acceptable but a fully correct answer of 2.6 was required for any credit. Many students, even some with a correct graph, clearly had no idea how to use their graph to answer this question.

Question 14

Many students gained the one mark for correctly describing the transformation as a reflection. Although a lot of students did not use the correct terminology instead using ‘mirrored’ or ‘flipped’ so getting no mark. However significantly fewer were able to fully describe it. Reflection in the line x or about the origin were common errors. Several appeared to refer to the x-axis as x = 0.

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Question 15

Most students gained at least one mark for a correct entry in the table in the ‘cotton fabric’ row, usually the first entry of £18; many then incorrectly completed this line as an arithmetic progression. Fewer students were able to deal with the ratio in order to gain any success for values in the ‘silk fibre’ row. Very few gained full marks on this question. It was rare to see working in this question; students should be reminded to show all working no matter how trivial.

Question 16

Both parts of this question were poorly answered. Confusion with the conversion between metres and centimetres prevented very many students completing a solution. Many found the volumes and but then divided the volume of a box by the volume of the van. A significant number of students correctly attempted to find the maximum number of boxes by finding the correct numbers, (6, 5 and 4), fitting in each dimension of the van. Unfortunately, many simply added these numbers (to get 15) or selected the lowest value and divided by 3 to find the time. Many students calculated the time by dividing the least number (in this case 4) by 3.

Some students calculated surface areas rather than volumes.

In part (b), very few students answered the question given. A correct response described how the time might change (greater or less) together with a coherent reason. Many students simply said that the time will change, without reason, or argued that there will be less boxes or it will be more difficult to fill the van without ever mentioning the time implication. Centres are advised to give their students more guidance on how to fully answer this type of questions.

Question 17

In part (a), the most common incorrect response was an answer of 16m.

Part (b) was poorly answered with many students seemingly selecting their choice at random.

Clearly the meaning of the words used in this question is an area that needs clarification for students.

Question 18

The predictable incorrect answer of n + 3 was the most common error made, in part (a). 4n + 3 was another incorrect answer seen. However, it was pleasing to see so many students scoring at least 1 for giving 3n as a part of their nth term, and often two marks.

Only a minority of students were able to use their nth term to solve part (b). It was more common for students to gain success by laboriously writing out the complete sequence up to 91. Many students incorrectly tried to substitute 90 into their nth term.

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Question 19

In part (a), many students simply selected the middle interval of the five intervals in the table; showing a lack of understanding of the concept of median in such a context.

In part (b), most students scored at least one mark for making at most one error in their diagram. It was common for plots to be at the end of the intervals and many students insisted upon joining their first and last plots. Quite a few students having accurately plotted the points failed to make any attempt to join them.

Many weaker students just drew a histogram. This gained no credit even when all heights were ‘correct’.

Question 20

Many students at this level found the demands of this question too great. Often students were able to make a start at a solution by quoting one correct conversion, usually 1.089 × 3.785 or 2.83 ÷ 1.46, but very few went further.

Many students incorrectly converted 108.9p to pounds with values of 1.89 being the most common error or simply £108.9. A great many did give New York as the most economical city for petrol but very few gave two correct comparable values. The failure to find two comparable values was very much linked to the fact that students didn’t really know what values they were finding when they were dividing or multiplying numbers, students should be encouraged to write their units with their calculations to help make sense of their answers.

Question 21

This question was very poorly answered with few students knowing the formula Volume = Mass/Density. The great majority either multiplied the mass by the density or divided density by the mass.

Again conversion between units was poor, many not knowing that 1 kg = 1000 g.

Question 22

Only a very few students understood the need to compare both ratios with a common element representing the green pens.

Triple ratios of 2 : 9 : 1 or 2 : 5 : 1 or 2 : 4 : 1 were common errors followed by divisions of 100 by 12, 9 or 7 Many students tried to use the two given ratios independently. Some candidates found a number of equivalent ratios, but didn’t stop once they had found a pair that would work.

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Question 23

In part (a), those students understanding the meaning of ‘reciprocal’ usually found the correct answer. Very many students did not. Fewer numbers of students understood the term ‘error interval’ in part (b). Some scored 1 mark for quoting either 9.75 or 9.85. Those few giving inequalities with the correct limits often made mistakes with the actual inequality signs chosen.

Question 24

Only a very small minority of students made any sensible algebraic attempt to solve this problem. Those that did, usually made errors when giving the algebraic lengths in the perimeter of the 8-sided shape. 70 divided by 8 was a common, incorrect starting point.

There were a few trial and improvement successes but usually this approach failed.

Question 25

The digits ‘7452’ were often seen and many students gained one mark for this. Often the answer was correctly given in standard form instead of as an ordinary number and often correct standard form was converted incorrectly; 0.00007452 and 74520000 were common errors.

Question 26

Both parts to this question were poorly answered. In part (a), clearly the demand for the best estimate for the drawing pin to land “point up” was a distractor as many students gave their reason for selecting Mel to be because she had the greatest number for the times the pin landed point up. This was not acceptable. A number of students felt the pin should have a 50% chance of landing point up, presumably because there were two outcomes, and selected Tom accordingly.

In part (b), the very few students that separately found the probability of the drawing pin landing point up and the probability of the drawing pin landing point down rarely multiplied them together to answer the question.

Question 27

The ability to solve a pair of simultaneous equations was only demonstrated by the few more able students, although many did attempt it. Of those who did make a correct start by multiplying one equation with the intention of eliminating one of the unknowns, often the method was flawed by more than one error. A common error was to use the wrong operation when attempting to eliminate a variable.

Trial and improvement methods were very rarely successful.

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Summary

Based on their performance in this paper, students should:

learn standard metric conversions, for example 1 litre = 1000 millilitres;

ensure they read each question carefully;

show working whenever a calculation is carried out, no matter how trivial;

learn the meaning of words such as ‘expression’, ‘identity and ‘inequality’;

learn all the necessary formulae for this tier of entry, including that for density

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Principal Examiner Feedback – Higher Paper 3

Introduction

There were some able students who made good attempts at most of the questions on the paper but many of the later questions were often not attempted because the majority of students were targeting the lower grades.

It was pleasing that many of the students showed clear steps of working on this calculator paper. Centres should continue to emphasise the need to show full working. It is often apparent that a calculation has been performed but unless values are correct, part marks cannot be awarded for processes not shown.

Some students lost marks through not reading questions with sufficient care. Errors were made because students made ‘assumptions’ rather than reading and using the information that was given to them.

Many students seemed unable to apply their mathematical knowledge to a situation they may not have previously met and did not recognise what was required. Questions that assessed problem solving techniques, even ones early in the paper such as 4 (application of ratios) and 6 (deriving and solving an equation), were poorly attempted.

Unnecessary early rounding or truncating, on this calculator paper, gave rise to inaccuracies.

Report on individual questions

Question 1

Part (a) was not answered particularly well. A common error was to choose the class interval 150 < h ≤ 160 which is in the middle of the five class intervals in the table. Some students chose the class interval 170 < h ≤ 180, often because 19 is the median of the five frequencies.

Part (b) was not answered as well as might have been expected considering that frequency polygons have featured regularly on examination papers for a number of years. Points were frequently plotted at the ends of the intervals or at the beginnings of the intervals rather than at the midpoints. Many students did join their points with line segments; some joined them with a curve and some did not join them at all. Some otherwise correct frequency polygons were spoilt by students joining the first point to the last point. A number of students drew histograms.

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Question 2

Many students did make some headway on this question and it was attempted in a variety of ways. Some students used one of the given costs to find a comparative value, for example changing 108.9p per litre into $6.01 per gallon which could then be compared with the cost in New York. Other students used both given costs to find comparative values, for example changing 108.9p per litre into £4.12 per gallon and changing $2.83 per gallon into £1.94 per gallon. Mistakes were often made with the conversions, for example dividing by 1.46 instead of multiplying by 1.46, and some students wrote 108.9p as £1.89. Students attempting to work out the cost (in $) per litre in New York often did the division the wrong way round. It might help students to clarify their thinking if they included units with their values, for example $2.83 per gallon, not just 2.83.

Those students that did find comparative values usually made the correct decision but it was common for students to make a decision without having found values that could be compared. A few students made the wrong decision despite having correct values. Students had to deal with two conversion factors and they did not always present a clear picture of what they were trying to do.

Question 3

Many students were unable to use volume = mass ÷ density. A very common error was to multiply the mass by the density in an attempt to find the volume and some students divided the density by the mass. Those that did use volume = mass ÷ density frequently gained only one mark because they gave no consideration to the units or because they dealt with the units incorrectly. Many students forgot to multiply by 1000.

Question 4

Students found this question challenging. In order to make progress they needed to associate corresponding parts from the two ratios in a way that would help them. Some students did write down a ratio equivalent to 2 : 5 and a ratio equivalent to 4 : 1 but often the components for green pens were not the same. Those that did write down the ratios 8 : 20 and 20 : 5 or the ratio 8 : 20 : 5 were often able to go on and work out the greatest possible number of red pens. Some students went from 8 : 20 : 5 to 16 : 40 : 10 and finally to 24 : 60 : 15, the last figure

being the number required, and a few went from 8 : 20 : 5 to calculate × 100.

Some partially correct ratios such as 8 : 20 : 4 were seen but these gained no credit. A very common error was for students to start the problem by adding the numbers in the ratios, working out 2 + 5 = 7 and 4 + 1 = 5.

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Question 5

In part (a) many students were unable to find the value of the reciprocal of 1.6 and a wide variety of incorrect responses were seen.

In part (b) one mark was often scored for showing 9.75 or 9.85 or both of these values but relatively few students went on to give a fully correct error interval. Those that attempted to write an error interval frequently made mistakes with the inequality signs. Some students used 9.84 rather than 9.85. Those that wrote the upper bound as 9.849 failed to indicate the recurring nature of the final digit.

Question 6

It was pleasing that some good attempts that used an algebraic approach were seen though these were relatively few in number. Some students wrote x and x + 7 on the diagram of the rectangle but made no further progress. Successful attempts generally started with students writing expressions for the lengths of the sides on the diagram of the 8-sided shape. The most common mistake at this stage was for the two lengths of 7 to be incorrect or missing. Students often went on to write down an expression for the perimeter of the shape and equate this to 70. Those that formed an equation were usually able to solve it correctly. The final mark was awarded if students used their value of x correctly to find the area of the 8-sided shape.

A common mistake was to multiply the area of one rectangle by 8 (number of sides). A few students mistakenly used 7x instead of x + 7. Trial and improvement approaches were seen; they were usually unsuccessful and so gained no marks.

Question 7

This question was well attempted. The main obstacle to a correct answer for many students was the inability to write 7.452 × 10–4 as an ordinary number. In many responses 7.452 × 10–4

was either converted incorrectly to an ordinary number or given as the final answer. Many students scored one mark for showing the digits 7452. Although the calculation is one that can be entered directly into a calculator there were a number of students who attempted, often unsuccessfully, to first write the numbers in the question as ordinary numbers.

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Question 8

Students who identified Mel in part (a) did not always give a correct reason. Some referred to her having the greatest number of points up, not to her having done the most trials. Tom was identified by some students because his results give the greatest probability of getting point up or because his results have the smallest difference between the number of points down and the number of points up.

In part (b) students were expected to use all the results to find the fractions and and

to then multiply these two fractions. Many students did find these two fractions. Those who

simplified them to and often went on to multiply but most students did not multiply the

two fractions. Many students added rather than multiplied. A few students used the results from just one of the three people but they could be awarded the method mark if they multiplied their two fractions to find the probability of point up followed by point down. Some students lost the accuracy mark because they gave the answer as 0.2. A decimal equivalent to a probability should be written to at least 2 decimal places (unless tenths).

Question 9

Part (a) was answered quite well. Students were often able to substitute at least one value of n into 12 500 × (0.85)n and gain the first mark. For some students this was just substituting n = 1 to get 10625. Some of those that showed enough correct further substitutions to answer the question chose the wrong number of years. Some used parts of a year, giving an answer such as 4.27, and could be awarded only one mark. A number of students just found 50% of 12500 (= 6250) and scored no marks.

Students were much less successful in part (b). Those who recognised that 79.20 = 60% of the interest before tax were sometimes able to work out the interest before tax. A common mistake was to work out 40% of 79.20 and then add the result to 79.20. Some students worked out the interest before tax as £132 but then stopped. Those that did attempt to work out 132 as a percentage of 5500 did not always complete the final step, giving an answer of 1.024 or 0.024 rather than 2.4. Some students started the question by working out 40% of 5500 as 3300. Although a small number of these students went on to give complete solutions most failed to make any further progress. Many students did not know how to start the question.

Question 10

In part (a) many students worked out the probability of getting a red counter as 0.05. A common incorrect answer was 0.5, often with 0.95 or 1 – 0.95 shown in the working.

Part (b) asked for the least possible number of counters in the bag. Students are advised to read the question carefully as a surprisingly large number gave a colour, not a number, as the

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answer. Sometimes they gave the lowest probability. Some students worked out the least possible number of counters as 20 but gave no reason for their answer; they scored one of the two marks. The most common correct reasons given referred to the numbers of counters having to be whole numbers. Some students gave a number greater than 20 as the least possible number of counters but scored one mark for a correct reason.Question 11

In part (a) many students correctly read the value of the median from the cumulative frequency graph. Common incorrect answers were 60, 38 and 55.

Correct answers were rarely seen in part (b). Most students either stated that Jamil is correct because the range is the largest value minus the smallest value or stated that he is incorrect because his calculation should have been 80 – 30 = 50. Very few students appreciated that the greatest value could be less than 80 or that the smallest value could be less than 40.

Many students gained one mark in part (c) for reading from the graph. This was usually done from a weight of 65 g and resulted in a cumulative frequency value of 48 or 49. The successful students subtracted this value from 60 to find the number of potatoes with a weight greater than 65 g and then either worked out this number of potatoes as a percentage of 60 or worked out 25% of 60 (15). A common error was failing to subtract the reading from 60. Some students got no further than reading from the graph.

Question 12

Working out 0.75 × 0.4 to get the probability of both spinners landing on white and working out 0.25 × 0.6 to get the probability of both spinners landing on red gained the two method marks. Some students found only one of these probabilities, usually the former, and scored one mark only. A number of those students who did work out both probabilities failed to spot that 0.3 is double 0.15 and therefore the answer will be double 24. Some students used 0.15 and 24 to work out that the total number of spins is 160 and were then often able to get the correct answer. A common error was to add the probabilities instead of multiplying them.

Question 13

Those students with some idea about completing the square were often able to score one mark for (x + 3)2 but errors were frequently made with the ‘– 16’.

Question 14

Relatively few students showed that they understood the relationships between lengths, areas and volumes in similar figures. Those who did recognise that they should use the ratio of the volumes, 27 : 8, to find the ratio of the lengths or the length scale factor were usually able to give a complete method to show that the surface area of cone B is 132 cm2. Many students assumed the result they were given instead of proving it.

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Question 15

In part (a) those students that gained the first mark by substituting two appropriate values into x3 + 7x – 5 often failed to make a deduction about the roots. Students seemed to think that getting one positive answer and one negative answer was sufficient. Many students had no idea how to show that the equation has a solution between x = 0 and x = 1. Attempts at using the quadratic equation formula were very common.

Many students were able to gain one mark in part (b) by showing a correct first step in the rearrangement, most commonly this was x3 + 7x = 5. Many, though, were then unable to continue with the rearrangement by using factorisation and show a complete method.

When answers were seen in part (c) it was evident that some students had a good appreciation of the process of iteration and they were able to gain the first method mark for substituting the starting value of 1 into the formula. When the results of the next two iterations were not accurate the second method mark could only be awarded if the substitutions were shown. Rounding or truncating the value of x2 resulted in some final answers that were not sufficiently accurate. Some students carried out more than three iterations. In these responses the accuracy mark was awarded for the value 0.6704 and any further iterations were ignored.

Part (d) was poorly answered. Those students who did gain one mark for substituting their answer to part (c) into x3 + 7x – 5 rarely compared the result of the substitution with zero to determine the accuracy of their estimate. Even when the correct value was substituted the result of the substitution was often incorrect.

Question 16

Most students failed to identify from the question that they needed to work with bounds and it was very common to see 11.8 and 148 substituted into the formula for petrol consumption. There were no marks for this approach which completely ignored the topic that the question was actually assessing. Identifying at least one upper bound or one lower bound was sufficient for the first mark. A few students used 148.49 instead of 148.5 but made no attempt to show that the 9 is recurring. Some students with the correct bounds did not appreciate that they needed to use the upper bound for litres of petrol and the lower bound for distance. Some substituted the two upper bounds or the two lower bounds into the formula. Those that did substitute the two correct bounds usually went on to make the correct decision.

Question 17

In order to start this question and work out the length of CD, students needed to recall that the

area of a triangle is given by ab sin C. Those that did recall this correctly often gained the

first mark for writing a correct statement such as 0.5 × 11 × CD × sin 105 = 56. Mistakes, though, were often made when rearranging to find CD. Those students that did not show a correct process to work out CD were still able to gain subsequent process marks: one mark for using the cosine rule to work out AC and one mark for using the sine rule to work out AB.

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A number of students assumed that the triangles were right-angled and tried to find CD by using base (AD) × height (CD) ÷ 2 = 56. It was not uncommon to see students attempting to use Pythagoras and SOHCAHTOA when trying to work out the length of AC.

Question 18

In part (a) relatively few students appreciated that working out an estimate for the distance the train travelled required them to find the area under the curve. Those that did usually showed 4 strips of equal width on the graph and made an attempt at working out the area. Some very good answers were seen but attempts were often spoilt by values being read incorrectly from the graph or by the formula for finding the area of a trapezium being used incorrectly. Some students worked with rectangles and triangles, often successfully. Many students, though, simply used distance, speed, time formulas and finished with wrong answers of 320 or 360.

Part (b) was only accessible to those students who had attempted to work out an area in part (a). Some students did state that their estimate was an overestimate and gave a reason linked to their method. However, many of the reasons given had nothing to do with the method used to work out the area.

Question 19

Some students rearranged the equation of the straight line to make either x or y the subject and those who realised that they needed to solve the equations simultaneously then substituted into the equation of the circle. Mistakes were often made when expanding the brackets and when simplifying the resulting quadratic equation. Students usually solved the quadratic equation to show that the line and the circle only intersect at one point although in a few responses the discriminant was used to show that there is only one solution. Some students solved the equation but made no concluding statement about how this proved that the straight line is a tangent to the circle and they were not awarded the final mark.

Question 20

Some of the students that started by drawing in the radius OC to give two isosceles triangles were able to go on and show that angle ACB = 90. Some did so by introducing algebraic notation whereas others used angle notation. It was pleasing to see some very good attempts but these were few in number. Students should note that this type of geometric proof does require full and correctly worded reasons to be given. It is not enough to state, for example, that angle OAC = angle OCA, it is also necessary to give a reason why. Those students that showed that angle ACB = 90 but gave no reasons or incomplete reasons were awarded three of the four marks. Some students ignored the statement “You must not use any circle theorems in your proof” and focused on angles in a semicircle.

Question 21

Some students scored one mark for AB = b – a or BA = a – b but few were able to make any further meaningful progress. Those that did were most likely to find a correct expression for

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MN. Few students wrote that AP = k(b – a ) which meant that correct expressions for MP and PN were rare. Mistakes were sometimes made with the direction signs of the vectors.

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Summary

Based on their performance in this paper, students should:

• read the information given in each question very carefully;

• practise solving problems that require an algebraic approach;

• use a calculator to work out accurate values without rounding or truncating early;

• practice using their knowledge in different ways and in a wide variety of contexts;

• recognise which rules and theorems are associated with triangles without a 90 angle and use them appropriately;

• give correctly worded reasons when presenting a geometric proof.

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1MA1 GCSE Mathematics (9–1)November 2017

1MA1 9 8 7 6 5 4 3 2 11F Foundation tier Paper 1F 44 35 25 16 7

2F Foundation tier Paper 2F 51 40 29 18 8

3F Foundation tier Paper 3F 49 38 28 18 8

1H Higher tier Paper 1H 64 51 38 29 20 11 6

2H Higher tier Paper 2H 62 49 37 28 18 11 7

3H Higher tier Paper 3H 62 49 37 28 19 10 5

(Marks for papers 1F, 2F, 3F, 1H, 2H and 3H are each out of 80.)

1MA1 overall 9 8 7 6 5 4 3 2 11MA1F Foundation tier 145 113 83 53 231MA1H Higher tier 189 150 112 85 58 32 19

(Marks for 1MA1F and 1MA1H are each out of 240.)

Grade boundaries are set by examiners for the whole qualification at 7, 4 and 1 and the intermediate grades are calculated arithmetically by calculating equal intervals. This might lead to some rounded figures and the thus three paper boundaries at a given grade may not add exactly to the overall boundary at that grade.

Thus, for example, the overall grade for 2 at Foundation tier falls a third of the way between 23 and 113 at 53. By the same token the grade 2 boundaries on each of the Foundation tier papers are strictly 16.33, 18.67 and 18 but for papers 1 and 2 those are rounded to the nearest whole number for the purposes of the table above.

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