Find Which GCSE Higher Maths Topics Are Essential to Success in The GCSE Higher Exams 2023

Higher Maths GCSE preparation this year is a challenge, given the past two academic years and how challenging they have been for the education sector, particularly in the domain of examinations (or lack thereof) and assessments.

Free predicted versions of GCSE maths past papers based on the Advanced Information provided by the exam boards. Choose from Edexcel, AQA and OCR higher and foundation papers. Answers and mark scheme included.

The current picture for higher maths

Maths study beyond GCSE

At GCSE level, a school-wide approach may be necessary, collaborating with colleagues to understand what skills are fundamental for success in post-16 study. This will depend on the school’s individual setting and demographic. For example, it may be important to ensure that statistical content (necessary for study of AS/A Level Psychology) is covered, or that students have sufficient algebra and graphing skills to be able to access A Level Sciences.

This blog focuses on strategies for exam preparation for the new Higher mathematics and draws upon research and analysis conducted on the six series of exam papers available from Edexcel (June 2017- November 2019), so please bear this in mind if you use AQA or OCR. This companion piece gives more detail and is my rationale for the suggestions below.

See also: How To Revise For GCSE

Number and proportion in higher maths

Key topics

• Basic calculation skills (four operations, working flexibly and accurately with fractions, decimals, percentages and ratio).
• Proportional reasoning, particularly ratio, assessed throughout the paper, and lack of understanding could limit accessibility of other topics.
• 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.
• Crossover between lower and upper bound calculations and compound measures, particularly speed and density.

Content to skim or skip

• Problem-solving or rich problems using HCF, LCM, product of primes, standard form, surds.

Frequently appearing questions 

For me, the key thing in preparing students for the Higher maths papers is ensuring that their basic calculation skills are absolutely watertight. There are significantly fewer marks in the marking scheme on Higher, for carrying out procedural arithmetic calculations – it is expected that students should be able to work quickly, accurately and flexibly with number, including calculating with negatives, fractions, decimals, percentages and ratio. Therefore, when these skills appear in other exam questions, it is crucial that these calculations can be carried out almost automatically.

Ratio needs a particular mention, because it appears throughout the Higher papers in a variety of contexts. The standard ‘split into a ratio’ or ‘find a quantity’ is not examined independently, but appears within other questions, often stating key information – for example, angles for a trigonometry problem given as a ratio rather than as values. Without the ability to carry out procedural ratio calculations, many questions become inaccessible. 

As with Foundation, there is a clear undercurrent of proportional reasoning throughout the papers, both in typical real-life problem solving contexts, and with more abstract concepts. Students are expected to be able to make connections between topics more readily, such as using exponential formulae to represent compound percentage change.

There are still some easy wins in the Number and Proportion strand; standard form, HCF and LCM, power calculations (particularly fractional indices), surd manipulation, expressing recurring decimals as fractions, using a calculator and error intervals are all topics assessed in a fairly procedural manner on most papers. More in-depth questions about lower and upper bounds tend to include context, often speed or density calculations.

In terms of exam adaptations, the formula sheet includes the algebraic formula for compound interest, but this is stated in formal terms, using “principal amount”, “interest rate” and the word “accrued”. It may be worth ensuring students are familiar with these terms if they intend to rely on the given formula in the exam.

Algebra in higher maths

Key topics

• Basic algebra skills (simplify, expand, factorise, solving equations and inequalities) need to be automatic. Algebraic fractions frequently used to assess a range of algebraic manipulation skills.
• Application of linear and quadratic equations and inequalities to a variety of contexts, particularly perimeter, area and volume.
• Basic graph skills, equation of a straight line, drawing graphs and estimating solutions from a graph.
• Procedural work within sequences and functions, such as quadratic nth term, inverse and composite functions.
• Simultaneous equations, including non-linear, expanding triple brackets, solutions of equations using iteration.
• Gradients and areas, particularly using speed-time and distance-time graphs.

Content to skim or skip

• More challenging multi-step coordinate geometry problems.
• Algebraic proof, as this is frequently not scaffolded.

Frequently appearing questions 

As with basic calculation skills, the algebra toolkit of simplifying, expanding and factorising is also assumed knowledge at Higher GCSE and is less likely to be assessed in a standalone question. Many of these skills are embedded within other problems; for example, algebraic fractions feature very highly, allowing for simultaneous assessment of expanding and factorising.

From past exams and specimen papers we can glean that students are much more likely to be asked to procedurally expand triple brackets rather than double brackets, because work on quadratic expressions and equations is more frequently assessed in context. Algebraic proof is generally very challenging due to the fact that many of the questions do not give any scaffolding or hints (such as the algebraic forms needed to complete the proof).

Students must be able to form linear and quadratic equations and inequalities from contexts, as well as being reliably able to solve using an appropriate method. Contexts include perimeter and area problems, nth term expressions and quadratic sequences, probability, and the trigonometric formula for area of a triangle, ½ ab sin C.

Candidates are likely to be asked to solve a system of simultaneous equations with one linear, one quadratic or other non-linear; this is also another opportunity to assess quadratic manipulation skills. As part of the 2022 adaptations, students are given the quadratic formula on the formula sheet.

Old maths higher past papers have included some very challenging coordinate geometry problems similar to those previously seen at AS Level. Given reduced teaching time, I wouldn’t focus on these in too much depth with the majority of students.

However, basic graph skills, such as drawing, using graphs to estimate solutions to equations, and finding equations of parallel and perpendicular lines are really important. There is some nice accessible content in non-linear graphs too, such as matching pictures of different graphs to their equations, or drawing graphs of quadratics and circles. 

A couple of other easy wins here include iteration, which (if it appears) is done in a step-by-step procedural manner, inverse and composite functions, and nth term of a quadratic sequence. It’s also worth mentioning that newer content on gradients and areas under curves seems to crop up quite a bit, usually in the context of speed-time or distance-time graphs, and, despite being considered a ‘harder topic’, is procedural and quite accessible.

Advance notice of topics may be particularly helpful with the algebra strand at Higher level, as the specification has a broad range of topics. From February, the advance information should hopefully allow us to narrow down to those topics or the GCSE maths questions we know will appear on the final papers.

Shape in higher maths

Key topics

• Perimeter, area and volume, particularly context-based problems.
• Compound measures, particularly speed and density.
• Pythagoras and trigonometry, with focus on problem-solving, double applications of rules, working in 3D or applying to area calculations.
• Clear mathematical reasoning for 2D shape and angle problems, particularly angles in polygons and circle theorems.

Content to skim or skip

• Constructions, scale drawings and bearings.
• 3D shape and angle properties.

Frequently appearing questions 

One thing I noted from the paper analysis is the large amount of problem-solving based around perimeter, area and volume; there were almost no procedural problems, in favour instead of real-life contexts and unfamiliar abstract situations. As well as ensuring students can reliably apply standard methods, they must be able to flexibly apply these skills to other problems.

There was significant crossover with solving equations and inequalities, both linear and quadratic, Pythagoras and trigonometry, and other 2D shape and angle reasoning. Another key area within perimeter, area and volume is using one measure to calculate another – for example, calculating the surface area of a shape given its volume.

Pythagoras and trigonometry was another key area. The majority of these questions had significant elements of problem solving. There was quite a bit of higher level trigonometry, such as sine and cosine rule problems, often requiring a double application, or use of Pythagoras, or some crossover with area calculations.

3D Pythagoras or trigonometry appeared in half of the series. As part of the 2022 exam adaptations, students are given all relevant Pythagoras and trigonometry formulae, therefore this may make questions on these topics slightly more accessible. Exact trigonometric values are not provided.

Speed and density featured heavily within work on units, measurements and drawings, while constructions, bearings and scale drawings appeared less frequently. While I don’t advocate skipping these topics completely, it’s perhaps worth just reviewing quickly before the exam period rather than going into considerable depth. Similarly, plans and elevations only appeared on a couple of papers, and questions were procedural and intuitive, so great depth is not required here either.

Probability and statistics in higher maths

Key topics

• Procedural work on combinations, Venn diagrams, tree diagrams.
• Presenting data: drawing and interpreting cumulative frequency graphs, box plots, histograms, frequency polygons.
• Estimating the mean and calculating averages and range from charts and graphs.

Content to skim or skip

• Data collection and sampling.
• Frequency trees.
• Problem solving using independent events and conditional probability.

Frequently appearing questions 

Like Foundation, a fair proportion of probability and statistics is assessed in procedural ways, although there are some challenging problems around independent events and conditional probability for the most able candidates. As such, I’d skim these with the majority of students. It is worth noting that the formula sheet gives candidates the addition rule for “or” probabilities: PA or B=PA+PB-P(A and B).

Procedural work on ‘newer’ topics such as Venns and combinations would be valuable, as these topics seem to appear frequently, and have so far been assessed in similar-looking questions. Tree diagrams featured exclusively over frequency trees (a direct reverse of the situation on Foundation), so frequency trees could be skimmed, plugging gaps once advance information has been released if necessary.

In statistics, the emphasis is more heavily on presenting data rather than processing data, with equal focus on drawing or completing charts and graphs, and interpreting or critiquing existing representations. Cumulative frequency graphs, box plots, frequency polygons appear frequently; scatter graphs usually include a drawing and interpreting component.

Processing data usually involves estimation of the mean from a table, or reading information from a pre-drawn chart or graph, such as calculating the interquartile range from a given box plot. Data collection and sampling doesn’t appear on every series, but this has usually been assessed in a standard ‘stratified sample-style’ question, and doesn’t need a great deal of depth.

Higher maths teaching strategies

Planning the content for a scheme of work for Year 11 will depend on how your particular setting and students have been impacted so far. For example, it might be prudent to begin by re-covering topics that many students may have missed or had patchy coverage of due to emergency closures. Beyond this, and in light of the proposed exam adaptations having no effect on the content examined, I don’t think there needs to be anything significantly different done for this cohort of students in terms of exam preparation. 

Making connections

One essential element of exam preparation for the Higher papers is to ensure that students are familiar with connections within the content. As already discussed, key topics here include ratio and percentages, linear and quadratic equations and inequalities, perimeter, area and volume, and Pythagoras and trigonometry.

Students need plenty of exposure to questions requiring applications of multiple topics, and plenty of opportunity to work on these problems in a collaborative way as a class. Many schemes of work devote most, if not all, of Year 11 to revision and exam preparation; this is an ideal time to explore links and show students the variety of contexts their skills can be applied to.

Developing resilience

It is also important to develop resilience and a willingness to have a go at most or all questions. We must remember that a significant proportion of the Higher paper is aimed at grade 7-9 candidates, and that some of the multi-step problems will be incredibly challenging for some candidates.

That said, we need to model and develop strategies for achieving maybe the first couple of marks on a 5-mark question – for example, in a multi-step problem involving a double application of trigonometry, maybe the student can get the first couple of marks for applying right-angled triangle trigonometry correctly. 

Low stakes practice

A careful balance needs to be struck in terms of assessment; students may not be ready to sit a formal mock exam and practice papers in the early Autumn term, as was standard practice in some schools pre-pandemic.

However, continual, low-stakes formative assessment (regular mini-quizzes, perhaps using the starting portion of the lesson) is more crucial than ever for identifying gaps in learning. GCSE intervention strategies need to be in place from September to pick up those students who are already struggling. 

Free GCSE maths revision resources for schools
In addition to the GCSE maths tuition we provide to secondary schools more support is available free for GCSE maths revision including:
– GCSE maths past papers
– GCSE maths worksheets
– GCSE maths questions

Third Space Learning offers one-to-one online intervention to help students prepare for the maths GCSE exam. Assessing the needs of each individual student, our intervention lessons are designed to plug the gaps in students’ learning and build confidence going into the GCSE maths exam.Third Space Learning also provides a wide variety of free resources and worksheets available on our website to help students with maths revision. 

What is right for the student

Finally, the decisions about Foundation vs the Higher maths exam for borderline grade 4-5 candidates may need to be made for a larger group of students. It is likely that there will be some students who, with the normal, pre-pandemic amount of teaching time and support from Years 9-11, would be encouraged towards taking the Higher paper to see if they got a 6, with a fallback of a grade 5.

However, given the disruptions to their learning, these students within the 2022 cohort may have considerable gaps in their learning, and it may be more appropriate to enter them for an exam paper where they can securely achieve a 5. It is worth bearing in mind that, with the newer papers, exam boards encouraged more Foundation entries, particularly those who would have previously been considered ‘weak grade Cs’.

Read more: GCSE grade boundaries

Third Space Learning’s GCSE online maths courses are designed to cover the entire GCSE maths curriculum. Each lesson provides step-by-step worked examples, multiple-choice practice questions designed to address misconceptions, practice exam questions with student-friendly mark schemes, learning checklists taken from the national curriculum and free, printable GCSE maths worksheets. By working through these lessons students are able to practise and know how to revise the GCSE mathematics content in order to develop mathematical fluency and increase their confidence. 

Read more:

• The Most Impactful GCSE Maths Topics
• Edexcel Maths Past Papers
• How to Revise: 20 Proven Revision Techniques

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