# Find Out What Topics You Should Be Teaching With This Question Level Analysis Of Edexcel GCSE Maths Higher Past Papers

**GCSE maths Higher past papers can be very useful in preparing students for the upcoming exams. The past two academic years have been very challenging for the education sector and lots of us are still trying to understand the implications for exam groups from 2022 onwards.**

**The initial motivation for this blog on Edexcel GCSE maths Higher past papers was as research for a**nother article on preparing Year 10 for Foundation GCSE, which raised a few interesting and surprising statistics.

Following this, I decided to have a similar in-depth look at the full set of six Edexcel GCSE maths Higher past papers from June 2017 through to November 2019. In this blog, I briefly discuss the methods used, then outline the interesting points for each series of papers. Finally I look at the full set of six papers, outlining common themes in each strand, and draw conclusions about implications for teaching.

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### Key findings in maths higher past papers

#### Number and proportion

- Basic calculation skills (four operations and working flexibly with fractions, decimals, percentages and ratio) need to be automatic.
- Proportional reasoning, particularly ratio, is assessed throughout the paper, and lack of understanding could limit accessibility of other topics.
- There is some accessible procedural work on standard form, HCF and LCM, negative and fractional indices, surd manipulation, expressing recurring decimals as fractions, using a calculator and error intervals, on most maths Higher past papers.

#### Algebra

- Basic algebra skills (simplify, expand, factorise, solving equations and inequalities) need to be automatic. Algebraic fractions are frequently used to assess a range of algebraic manipulation skills.
- Linear and quadratic equations and inequalities are frequently assessed in a variety of contexts, particularly perimeter, area and volume.
- There is some accessible procedural work within sequences, functions, such as quadratic nth term, inverse and composite functions, basic graph skills, equation of a straight line, drawing graphs and estimating solutions from a graph.
- Other key topics include: simultaneous equations, including non-linear, expanding triple brackets, solutions of equations using iteration, gradients and areas, particularly using speed-time and distance-time graphs.

#### Shape

- Perimeter, area and volume is almost exclusively assessed in context and links heavily to other topics, such as linear and quadratic equations and inequalities.
- There is increased emphasis on compound measures, particularly speed and density.
- Pythagoras and trigonometry has a stronger focus on problem-solving/multi-step problems requiring applications of more than one rule, working in 3D or applying to area calculations.
- Candidates need to be able to give clear mathematical reasoning for 2D shape and angle problems; angles in polygons and circle theorems feature here.

#### Probability and statistics

- There is plenty of accessible procedural work on combinations, Venn diagrams and tree diagrams.
- Presenting data mostly involves drawing and interpreting cumulative frequency graphs, box plots, histograms, frequency polygons; candidates were just as likely to be asked to interpret or critique a given chart or graph as to draw one themselves.
- Processing data focuses on estimating the mean, and calculating averages and ranges from charts and graphs (such as finding the IQR from a box plot).

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### Methodology for categorising maths higher past papers

#### Categorising maths higher past papers by topic

One thing I found really challenging about this series of blogs was getting the level of detail right. When I began my analysis of maths past papers for the Foundation blog, I used a fairly broad approach when categorising each question, drilling down further into particularly interesting topics or those which attracted a large number of marks. When looking at the Higher tier maths series, I used the same categories, but also noted down a few key details for each question; this was necessary for Higher level maths, due to an increased number of topics, and also more questions with applications of multiple topics.

A few years ago, as part of our GCSE scheme planning for the changes in 2015, I took the KS3 and 4 Programmes of Study and divided them into ten key areas, with each area split further into topics and sub-topics. We used this for our scheme of work, and I still use this now to organise my electronic resources. I decided to use this structure for my analysis, as it’s already closely aligned with the categorisation that the exam boards use. Content in bold is examined at Higher level only.

As for my Foundation analysis, classifying some of the multi-mark, deeper problem-solving questions presented a bit of a challenge; I decided to pick a main topic for each question and classify under that heading, rather than try and assign marks for topics within questions. So, for example, June 2017 P1 Q18 required candidates to apply knowledge of properties of a rhombus to a question about linear graphs; however, as the question is inaccessible without knowledge about the general equation of a straight line, I classified this as ‘linear graphs’.

It should be noted that the proportional reasoning category only covers standard proportion and ratio-type problems – so recipes, conversions, splitting into ratios and so on. However, many ratio problems do include fractions, percentages or both.

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#### Categorising maths higher past papers by complexity

In order to simplify the analysis process, I decided to classify the high-mark problem-solving questions in the maths Higher past papers by their main topic; this does lose a little bit of the detail, but does make it easier to look for trends across a larger data set.

I also wanted an idea of the difficulty or complexity of each question; I decided to base this broadly on the GCSE assessment objectives:

- A01 – using and applying standard techniques (40%);
- A02 – reasoning, interpreting and communicating mathematically (30%);
- A03 – solving problems in mathematics and other contexts (30%).

It should be noted that this is a very rough basis, as longer or multi-step questions often award marks for more than one assessment objective in the mark scheme. My complexity classifications can be stated as follows:

- C1 – standard procedural problems, often worth one or two marks;
- C2 – problems asking candidates to explain their reasoning, interpret information from tables, diagrams or other contexts, or ‘show that’ type questions;
- C3 – non-standard multi-step problems, often requiring the application of skills from a variety of topics, worth three, four, or more marks.

This is also not a ‘difficulty’ measure – it should be noted that ‘higher grade’ topics, such as simultaneous equations or estimating the mean, are often examined as standard procedural.

Apart from June 2017, roughly half of the marks were allocated to C3 problems; obviously within this there will be some A01 marks for applying standard techniques. However, this emphasises the expectation that Higher maths candidates should be able to reason flexibly with their mathematical knowledge, rather than just apply to standard problems. From the six series of maths Higher past papers analysed, the November series seem to favour a slightly higher ratio of C2 to C1 questions, particularly in the last two years that we have exam papers for.

### Notes on the maths higher past papers

As I began working through the GCSE maths Higher past papers and compiling the data, I also noted some features of each set of specimen papers. I didn’t really have a hard-and-fast method here; these were just the things that jumped out at me once I’d completed the data crunch of each paper set.

#### June 2017 maths higher past paper

- Number and proportion 21%
- Shape 34%
- Algebra 30%
- Probability and statistics 15%

As the first series of maths Higher past papers for the most recent GCSE specification, this set feels challenging. Issues with the difficulty with the back end of the paper was addressed in the Chief Examiner’s Report in June 2018. This series hit a significant number of the newer topics, and a large proportion of what would have been considered grade A/A* material on the preceding syllabus and in older maths Higher past papers. There is a considerable skew towards shape and algebra; number content feels quite light, particularly assessed in an explicit manner.

**Number and proportion in the maths higher past paper**

Compared to the Foundation paper, there is significantly less number work, particularly arithmetic, in the maths Higher past papers. Of the 19 marks available, 7 of these were for standard form (a mixture of C1 and C2 questions) and 7 for working with fractional indices, including one negative.

There was also significantly less work explicitly on proportional reasoning than on Foundation, although there were two C3 5 mark questions (P2Q2 and P3Q4) requiring candidates to solve problems using fractions, ratio and percentages. The remaining 4 marks were for a fairly standard C1 inverse proportion equation – write the equation, then use this to find a value.

**Algebra in the maths higher past paper**

Unlike Foundation, candidates were expected to do substantially more problem-solving using algebra.

Linear and non-linear graphs featured quite highly on this series, with 10 and 15 marks respectively; together, these accounted for nearly 35% of the algebra content. P1Q18 required candidates to use knowledge about properties of a rhombus to solve a problem about perpendicular lines, and P1Q6 asked candidates to show that two lines are parallel. There was a nice accessible C1 question on non-linear graphs requiring candidates to match the graph to the equation (P2Q14).

Algebraic fractions was hit hard, with 11 marks in total (P2Q11, Q19 and P3Q14) – two questions on manipulation and one solving an equation involving algebraic fractions. The latter was the only question categorised purely as ‘linear equations’; this content is assessed in contexts in the rest of the paper.

There were plenty of really challenging algebra questions; see P2Q23 for a good example, involving finding the tangent to a circle, complicated further by inclusion of a surd coordinate.

Quadratic equations were mostly assessed in context (see P3Q15 where candidates had to apply the trigonometric area formula for a triangle to derive and then solve a quadratic using the formula). There was however one C1 quadratic inequality, and a triple bracket expansion.

**Shape in the maths higher past paper**

A third of the marks available for shape were on questions involving Pythagoras and trigonometry; as discussed above, some of these questions pulled elements from other topics, but were inaccessible without some application of Pythagoras or trigonometry. A significant proportion of these were C3 – of the 27 marks available, 25 were split across 5 questions, each worth 5 marks. These included using circle theorems and Pythagoras to solve a problem about arc length (P3Q18) and using the trigonometric formula for area of a triangle to find the area of a shaded section within a sector (P2Q17).

All 14 of the marks for units, measurements and drawing were for compound measures (speed and density). The majority of these were C3, with some C2 asking candidates to explain how assumptions would affect their answers.

Transformations only appeared once; a 3 mark C3 question which was very wordy.

I also noted that quite a few of the ‘show that’ type questions, particularly in 2D shape and angle properties, ask candidates to ‘prove’, requiring more rigour in their presented solutions.

**Probability and statistics in the maths higher past paper**

Within probability, quite a lot of the newer content (Venns, combinations, probability notation) featured. P2Q12 on tree diagrams required candidates to describe what was wrong and then answer a follow-up question requiring a good understanding of the independence law. However, all of the probability content was C1 or C2.

It felt like there was comparatively little coverage of statistics content on this series; within presenting data, scatter graphs, box plots, cumulative frequency and histograms featured. Candidates were more likely to be asked to estimate from a pre-drawn graph or chart than draw their own. All of the presenting data content was C1 or C2, with the only C3 question being P1Q7 involving problem solving using the mean.

#### November 2017 maths higher past paper

- Number and proportion 23%
- Shape 27%
- Algebra 32%
- probability and statistics 18%

**Number and proportion in the maths higher past paper**

Within the number strand, there were some fairly accessible C1 procedural questions on the non-calculator question paper, such as fractional indices (P1Q10) for 3 marks, a simple prime factorisation (P1Q1), ordering recurring decimals (P1Q8) and writing a recurring decimal as fraction (P1Q15). On the calculator papers, C1 questions included using a calculator and rounding the answer (P2Q8), writing an error interval (P3Q5b) and a couple of more straightforward percentage questions.

However, within the fractions, decimals, percentages and proportion strands, the majority of the questions were C3 context-based problems; even the question on exchange rates (P3Q2) required multiple steps and was quite challenging for early in the paper. It’s also important to be aware of the greater emphasis on links between repeated compound change and exponential formulae, as in P3Q9, a compound depreciation question.

**Algebra in the maths higher past paper**

50% of the algebra content was on sequences, functions and graphs, with 28 marks for graph work. Of the 11 marks available for linear graphs, all were C3 except 3 marks for a shading regions question on inequalities – which, despite being C1, was challenging because of the topic. The same was true of the non-linear graphs work; all 10 marks available here were C2 or C3. Many of the graph problems required application of other content or contexts – for example, P1Q19 asked candidates to find the equation of a straight line using a perpendicular line within the context of a rectangle drawn on a graph.

Again, most work on solving equations, both linear and quadratic, is in context. For example, P1Q23 required candidates to derive, then solve a quadratic inequality from a context about areas of triangles and rectangles, while P3Q6 required candidates to derive a much more simple linear equation in a problem about perimeter and area of rectangles.

**Shape in the maths higher past paper**

This series was slightly more balanced between the two shape strands, but over 44% of the marks for shape topics came from perimeter, area and volume, and Pythagoras and trigonometry, and all of this, bar one question, was C3. Some examples of these sorts of problems include P3Q17, quite a tricky problem using the trigonometric formula for area of a triangle and sine rule, and P2Q20 involving sector area, circle theorems and trigonometry.

A further 11 marks were available for 2D shape and angle properties work; these were mainly C2, requiring candidates to clearly explain their reasoning, with one C3 question on interior angles (N17P2).

**Probability and statistics in the maths higher past paper**

The probability and statistics weighting was a little more balanced on this series of maths Higher past papers. There were plenty of C1 questions in the presenting data strand, including drawing box plots, histograms and frequency polygons (11 marks in total). P1Q5a was an accessible C1 mean estimation for 3 marks. Sampling appeared on this series (P1Q13a), looking at a proportionate/stratified sample. There were also plenty of C2 questions requiring candidates to comment on their answers to previous questions.

Tree diagrams, independent and conditional events appeared quite a lot, with some C3 problems – for example, P2Q21a could use a tree diagram in its solution, but no scaffolding is given to point candidates towards this.

#### June 2018 maths higher past paper

- Number and proportion 21%
- Shape 30%
- Algebra 33%
- Probability and statistics 16%

**Number and proportion in the maths higher past paper**

There were 6 C1 marks available for finding HCF and LCM, including some in simple contexts, and 4 for adding or dividing mixed numbers. All of the work on powers and roots was C1 or C2, with questions on fractional and negative indices (P1Q9) and spotting mistakes in someone’s work on rationalising the denominator (P2Q20).

While the amount of proportion and ratio content assessed on its own seems low (5%), ratio in particular is embedded in questions on different topics, such as P3Q13, where the surface area of two shapes is given in a ratio, and candidates are then required to find the volume of the smaller shape.

**Algebra in the maths higher past paper**

Some of the algebra on this series felt more accessible than the previous two. All non-linear graph work, unusually, was C1; this did include some more challenging topics, such as equation of a circle (P2Q16), but there were nicer questions requiring candidates to match non-linear graphs to their equations, and draw a graph of a quadratic function (P2Q5). There were 4 marks available on linear real-life graphs, a mixture of C1 and C2, 6 marks on a step-by-step iteration question (P3Q18), and 11 C1 marks within algebraic manipulation, including laws of indices (P2Q1) and algebraic fractions (P1Q17).

However, as with the previous two series, equations were mostly assessed in context, such as P3Q16a, requiring candidates to derive and solve a pair of simultaneous equations from information given on a quadratic sequence.

**Shape in the maths higher past paper**

Again, perimeter, area and volume, Pythagoras and trigonometry, and 2D shape and angle properties carried most of the marks (almost 62% of the total shape content). 11 of the 17 marks in the Pythagoras and trigonometry strand were over 3 C3 questions, and brought in elements of other topics, such as P3Q19, which used right angled triangle trigonometry to derive a quadratic to solve. There were 8 marks over two questions (P2Q19 and P3Q13) which required candidates to find surface area given volume or vice versa.

Compound units featured highly, with speed, pressure and density all appearing, and all in C3 questions. The question on density (P2Q21) also included quite a complex bounds calculation.

**Probability and statistics in the maths higher past paper**

The statistics content on this series seemed particularly low (5%); all C1 or C2, including drawing and interpreting a box plot (P1Q10) and histogram (P2Q17). The only question on processing data was estimating the lower quartile from a histogram.

18 of the 19 marks on probability basics were C3, including a couple of tricky questions requiring candidates to work out the type of diagram needed (P2Q8 two-way table, P3Q20 Venn). P1Q16 required candidates to work with ratios to solve a problem about counters in a bag.

#### November 2018 in the maths higher past paper

- Number and proportion 21%
- Shape 34%
- Algebra 32%
- Probability and statistics 14%

**Number and proportion in the maths higher past paper**

This series had relatively few C1 procedural number questions, with nothing under factors, multiples and primes or fraction calculations, and very little on standard form or four operations. Nearly 41% of the number content was examined on percentage questions, with 9 of the 15 marks available were on multi-step C3 arithmetic problems appearing towards the start of the papers. While accessible for most candidates, they do require some careful thought early on, and it’s important to ensure marks aren’t thrown away here. P1Q11 was an interesting non-standard problem about percentage changes and proportion.

P1Q16 was a standard recurring decimal to fraction problem, but asked candidates to ‘prove algebraically’, so worth bearing in mind as this may put some off.

**Algebra in the maths higher past paper**

Of the 18 marks for algebraic manipulation, 13 of these involved algebraic fractions, including one with difference of two squares (P2Q12), and a tricky C3 problem which required candidates to use their knowledge of conditional probability to write algebraic fractions, get quadratic and then solve (P1Q22).

Again, lots of the work on equations appeared in contexts or required some work before solving, such as P2Q17, which required candidates to form and solve a pair of simultaneous equations with information given as ratios for 5 marks. Unusually, P3Q9c was a straightforward C1 question on use of the quadratic formula for an accessible 3 marks. P1Q17 required candidates to complete the square and find key points of a quadratic function to then sketch the graph; challenging but procedurally straightforward.

**Shape in the maths higher past paper**

Nearly 35% of the shape content was perimeter, area and volume, all C3 and mostly 4 or 5 mark questions. Some of these appeared relatively early in the maths Higher past paper; for example, P2Q5 was a 5 mark question using area of a trapezium and volume of a cylinder in context. As with previous series, there was considerable crossover with other strands, such as P3Q16, requiring an application of trigonometry as well as arc length.

As mentioned before, information in some questions was presented in ratio form; two key examples include P3Q6, a standard right angle triangle trigonometry problem, but with angle sizes given in a ratio, and P1Q15, a similar area and volume problem with the volume link stated as a ratio.

There were 13 marks on speed calculations or problems, including another C3 question (P3Q18) on bounds worth 5 marks. Density was also examined (P2Q20) in a question on volume of a frustum.

**Probability and statistics in the maths higher past paper**

The balance on this paper swung in favour of statistics, with very little probability content assessed explicitly. P2Q1 required candidates to complete a Venn diagram, then use this to find the probability that an element appeared in the union; this format of question appeared on at least two other series (J17P3Q1 and J19P3Q1). There was very little on independent and conditional probability, and no tree diagrams.

Of the 15 marks for presenting data, 10 of these were for commenting, interpreting or using pre-drawn charts and graphs, such as drawing a box plot from a given cumulative frequency graph (P1Q9), or commenting on whether statements about the graphs were correct.

#### June 2019 maths higher past paper

- Number and proportion 27%
- Shape 28%
- Algebra 27%
- Data 18%

**Number and proportion in the maths higher past paper**

The number content seemed fairly well-spread on this series, with C1 questions on HCF, use of calculator, multiplying mixed numbers, standard form, and a straightforward compound interest calculation (P3Q2). There was an interesting little C3 1 mark question on powers of 10 (P1Q8b), and a denominator to rationalise with tricky powers (P1Q18b).

The amount of simple proportion and ratio problems on this series was relatively high (10%). There were 12 marks on proportion, including a straightforward C1 recipe question (P1Q2) and combining proportion equations (P1Q20). Of the 12 marks available for ratio, all were context-based problem solving, and mostly C3 questions.

**Algebra in the maths higher past paper**

Unusually, there were 13 marks on fairly obvious quadratic expressions and equations, including completing the square and finding turning point (P1Q19) and expanding triple brackets (P3Q18). P1Q17 is another good example of ratios being embedded into all sorts of topics, as candidates were asked to form and solve a quadratic equation from a ratio.

There was slightly less work on non-linear graphs than on previous series, but 7 marks on gradients and areas (P2Q14). There were also 10 marks on sequences and functions, including a 5-mark C3 problem involving composite functions (P1Q21)

**Shape in the maths higher past paper**

In a similar vein to the previous series, a large proportion of shape marks were allocated to perimeter, area and volume, and Pythagoras and trigonometry. As before, there is considerable crossover between other topics, such as a non-calculator multiplication of fractions in a question on exact trigonometric values (P1Q14), and ratio and percentages in a problem about circumference and area (P2Q9). All of the perimeter, area and volume, and nearly all of the Pythagoras and trigonometry was C3.

There was very little on compound units compared to other series – only P3Q13 on density, and nothing on speed. There were 6 marks available on vectors (P2Q20), 2 marks for expressing as a vector (C1) and then a follow-up 4 marks on a C3 problem-solving question with ratio.

**Probability and statistics in the maths higher past paper**

Within the probability strand, there were 5 marks available (P3Q1) for completing a Venn diagram and finding the probability that an element is in the intersection – as previously mentioned, this question has appeared on two other series so far. Combinations also appeared for a total of 4 marks. Independent and conditional events featured highly, with 10 marks across two questions – one standard C1 completing a tree diagram and then using this to calculate a probability (P2Q10) and the other a significantly more challenging C3 problem (P1Q22).

Presenting data looked at time series, frequency polygons, cumulative frequency graphs and an interesting question requiring candidates to compare a histogram and a pie chart (P3Q21). The question on frequency polygons (P3Q3) with a follow-up to find the class interval containing the median was incredibly similar to November 2017 P3Q1. Sampling appeared on this series for 3 marks, and is again extremely similar to the sampling question on the November 2017 series.

#### November 2019 maths higher past paper

- Number and proportion 24%
- Shape 32%
- Algebra 27%
- Data 17%

**Number and proportion in the maths higher past paper**

As with the June 2019 series, there was a decent spread of accessible C1 questions within the number strand, including use of calculator, multiplying mixed numbers, writing a recurring decimal as a fraction, and standard form conversions. P1Q16 was a procedurally straightforward 5 marks on surds (rationalising the denominator).

The 6 marks on proportion were all on inverse proportion – 3 marks for a C3 inverse proportion problem (P2Q8), and 3 marks for a more straightforward C1 inverse proportion equation (P3Q16). As with June 2019, the 12 marks available for ratio were all C3, split evenly over three questions.

**Algebra in the maths higher past paper**

Most equations work again appeared in context, with forming and solving quadratics using composite functions (P1Q8b) and from two nth term rules (P2Q6). P1Q21 asked candidates to sketch the graph of a quadratic function and mark its key points. Quadratic inequalities also appeared, with a very challenging question requiring candidates to use ½ absin C to derive a given quadratic inequality for 3 marks (P2Q23a). However, as the inequality was given, solving in part (b) was procedural for a further 3 marks.

Within sequences and functions, there were a couple of nice C1 questions: P1Q18 asked candidates to find an inverse function, and P3Q20 involved finding the nth term of a quadratic sequence.

**Shape in the maths higher past paper**

Compared to some of the previous series, there was less on perimeter, area and volume and slightly more on units, measurements and drawings. P1Q8 was an interesting question about bearings and scale drawing, using a speed calculation. Quadratic equations were embedded again in P1Q14, which required candidates to form and solve quadratic from a context involving the area of a triangle.

Nearly 20% of the shape marks were for 2D shape and angle properties, including 7 marks on circle theorems. P3Q8 was a challenging C3 problem involving interior angles in polygons and P1Q5 on angle sum in a triangle included question information given as a ratio.

**Probability and statistics in the maths higher past paper**

There was a heavier skew in favour of statistics on this series, particularly in presenting data. Of the 19 marks available, 9 were for procedural C1 questions involving drawing or using cumulative frequency graphs or box plots. A further 7 were C2 marks for analysing, comparing or criticising graphs and charts. P2Q18 asked candidates to complete a histogram, but required some non-standard problem solving. Sampling appeared again (P2Q4) in an almost identical way to previous series (see June P2Q3).

Most of the marks in probability were for work on independent and conditional events, including one C1 tree diagram (P3Q11) and two more complicated C3 questions (P2Q16 and P3Q11).

### Considering complexity and implications for teaching

Looking at the six series of maths Higher past papers as a whole reveals some interesting patterns about the types of questions set and the complexity distribution for each topic. I’ve split the data set out into the four strands, Number and Proportion, Algebra, Shape and Probability and Statistics, and added commentary below.

#### Number and proportion in the maths higher past papers

The first thing I noticed was the dramatic degree of skew between C1 and C3 for some of the topics within Number and Proportion. For example, 67% of the marks for the Properties and Calculations strand were for procedural C1 questions, while 78% of the marks for Proportional Reasoning were C3.

Place value and ordering was almost all standard form questions, with plenty of accessible C1 marks requiring candidates to convert to and from standard form, or carry out standard form calculations. C2 marks in this topic were for ‘show that’ type questions, spotting errors in presented working, or for explaining how a change to the calculation would affect the answer.

Rounding and estimation had a mix of types of rounding, but error intervals accounted for a third of the marks and appeared on nearly every series. There were a handful of rounding to decimal places and significant figures, usually as a follow-up to a use of calculator problem. Questions involving lower and upper bounds were much more likely to be assessed in context through other topics, such as compound measures.

Four operations, factors, multiples and primes, and fraction calculations were all C1, although as noted above, these skills are also embedded throughout the maths Higher past papers. All of the fraction arithmetic questions involved mixed numbers. There was a mix of prime factorisation, HCF and LCM, and most of the work on four operations was using a calculator.

There was an average of nearly 4 marks per paper on powers and roots; 57% of these were C1, and included working with fractional and negative indices, simple laws of indices and rationalising the denominator in surd questions. The C2 questions (18%) within this topic usually asked candidates to spot mistakes in a solution. In decimals, all of the questions bar one involved working with recurring decimals – either ordering or writing as a fraction. The C2 questions were ‘show that’ or ‘prove’.

Within the remaining topics, the skew is more in favour of C3 marks: percentages (70%), proportion (60%) and ratio (98%), asking candidates to reason in unfamiliar, non-procedural ways, or to solve multi-step problems. For example, there were no questions asking candidates to simply ‘split into a ratio’ – this was assessed through other topics by presenting key question information as a ratio. Key topics in percentages were compound interest and change, reverse percentages, percentage profit and use of exponential notation to represent repeated change. Most of these were presented in real-life contexts, such as depreciation of price or money in a bank account.

#### Algebra in the maths higher past paper

Algebra attracts a fair proportion of C1 and C3 marks, with relatively few C2. As expected, the applications topics, such as quadratic equations and graphs, have a larger proportion of C3 marks.

Unlike Foundation, algebraic manipulation is only 48% C1 procedural questions; candidates are expected to be able to reason and solve problems using their algebra toolkit. Just over 51% of the C1 marks were on algebraic fractions; it’s worth noting that these questions also included things like expanding and factorising, including difference of two squares, so we can see that these basic skills are being assessed more commonly through work on algebraic fractions, rather than as standalone questions, as on Foundation.

There was a decent proportion of C1 marks available for the following topics: quadratic expressions (76%), simultaneous equations (75%), gradients and areas (72%) and sequences and functions (67%). However, these are typically considered more difficult topics. Candidates were three times more likely to be asked to expand triple brackets than double brackets, and of the C1 simultaneous equations questions, just over 55% of these were for one linear, one quadratic.

The questions on iteration were done step-by-step, were generally procedural and followed a set pattern, so could be easier marks to pick up. This appeared in November 2017 and June 2018, with nothing on the 2019 series.

Note that linear and quadratic equations were frequently examined in other C3 contexts, particularly perimeter, area and volume or shape and angle properties.

Graph work, both linear and non-linear, had a fair proportion of C3 problems; these were generally multi-step coordinate geometry problems, similar to those seen at early A Level, rather than applications to other topics. That said, 53% of the marks on non-linear graphs were available on C1 questions, including drawing graphs of quadratics and circles, matching graphs with equations (this style of question appeared on half of the series so far), and estimating solutions to equations from graphs.

#### Shape in the maths higher past paper

Similarly to the Foundation papers, in the maths Higher past papers there was a heavy emphasis on C3 applications within perimeter, area and volume (92%), and greater focus on reasoning with speed, distance and time within the units, measurements and drawing topic, linking to another context for proportional reasoning. I was also surprised at the high proportion of C3 problems on Pythagoras and trigonometry (88%).

Of the 65% of the C3 marks within units, measurements and drawings, quite a few of these were allocated to four- or five-mark questions, and most involved compound unit calculations. Speed featured the most, closely followed by density, and some of these problems also required bounds calculations (see June 17 P2Q21).

Candidates were unlikely to be presented with a simple right-angled triangle problem to solve using Pythagoras or trigonometry. Questions involving simple applications of trigonometry often had question information presented in a non-standard way, such as giving angles as ratios rather than values. Higher level trigonometry, such as sine and cosine rule problems were almost exclusively C3, and often required a double application, or use of Pythagoras, or some crossover with area calculations. 3D Pythagoras or trigonometry appeared on half of the series.

Perimeter, area and volume was nearly all assessed in an applied manner, with some series having all of this topic assigned C3. There were fewer real-life contexts than on Foundation, but problems requiring multiple steps, such as calculating a volume given a surface area, or vice versa, featured highly. There are very few simple questions asking candidates to calculate an area or volume, possibly because many formulae, particularly for volume and surface area, are given.

Transformations had a Higher percentage of C1 marks available (46%); enlargements appeared more frequently than any of the other three transformations. The C2 marks within this topic were mostly for describing transformations, so were also fairly accessible.

2D shape and angle properties attracted the highest proportions of C2 marks (46%), purely because many of the questions in this topic contain the statement: ‘give reasons for your answer’. The most commonly assessed topics here were circle theorems and interior angles in polygons.

#### Probability and statistics in the maths higher past paper

Over the six series of maths Higher past papers, the ‘new content’ (i.e. added in 2015) is strongly represented in the Probability strand. Unlike Foundation, there were no questions with pre-drawn frequency trees (although candidates could have chosen to use these for non-structured questions), with focus almost exclusively on tree diagrams. Venn diagrams and combinations featured highly, appearing on most series.

There was a fair split of procedural C1 and more challenging C3 problems in most of the Probability strand – C3 questions included a non-structured problem lending itself well to Venn diagrams (June 18 P3Q20), and a problem requiring ratio calculations in addition to probability (June 18 P1Q16).

As with Foundation, the majority of statistics content is C1 and C2, although there is slightly more problem-solving expected at Higher. Presenting data and processing data have only 11% and 24% respectively allocated to C3 marks. Furthermore, just over 70% of the statistics marks were for presenting data. A large proportion (43%) of presenting data is C2, meaning that candidates are almost as likely to be asked to analyse, read information from or critique a graph or chart as they are to draw one. Cumulative frequency charts, box plots, histograms and frequency polygons appeared most frequently, with some work on time series graphs and pie charts.

In processing data, quite a high proportion (52%) was C2 – this tends to be reading or estimating averages or range from given tables or charts. In contrast, only 24% of the marks for processing data were C1, and these were almost exclusively for estimating the mean from a table. Sampling was assessed on half of the series so far; it appeared in June 2019 (P2Q3) and November 2019 (P2Q4) in an almost identical format with a ‘stratified sample-style’ question.

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