Our best value package on a cost per hour basis, the 'Twenty Session One to One TMUA Coaching Package' is designed for students who want to thoroughly prepare for the TMUA at a steady, comprehensive pace. It suits the following: 

• students who wish to build confidence and skills from the ground up guided and coached all the way by a TMUA expert, in order to significantly improve their TMUA score; 
• students who want to study independently for the exam but also want to have an expert TMUA coach on hand as and when they need them, to guide them through certain less familiar areas of the syllabus or refine TMUA answering techniques; and
• students who are already fairly close to exam-ready and want to further polish every aspect of their TMUA readiness to achieve top mark potential.

Across twenty sessions, the student works closely with our expert TMUA coach to dissect and analyse hundreds of past TMUA questions, covering in depth either the entirety of topics that typically appear on the TMUA exam, or specific topics requested by the student (depending on the students' wishes). The sessions focus on developing robust problem-solving strategies, honing exam techniques, and filling any mathematical or logical gaps, ensuring the student approaches the TMUA fully prepared and capable of maximising their score.

Feel free to contact us to book a free no-obligation consultation to discuss your TMUA options with us.
For more general information on our online sessions check out our  'Online Coaching FAQ'. For session pricing and payment options, check out our Pricing and Payment page.

Note: We recommend booking early to ensure a steady pace of sessions leading up to exam readiness, avoiding pre-exam panic cramming!

IMPORTANT NOTE: Gresty Academy provides independent coaching and academic support for TMUA students and examination candidates. We are not affiliated with, endorsed by, or approved by UAT-UK, Pearson VUE, the University of Cambridge, the University of Oxford, or any other participating university or professional body. Our coaching is designed to supplement students’ own preparation for their respective examinations. As success in examinations depends primarily on individual student effort, discipline, revision and performance, Gresty Academy cannot and does not promise or guarantee any specific exam score, grade attainment or grade improvements.

Session #1: TMUA Prerequisite Maths One

Session Length: 60 minutes. Most students taking TMUA will also be taking A-Levels in Maths, IB in Maths and/or Further Maths A Level. As such, TMUA assumes knowledge of pretty much all maths up to A-Level, including maths that may have been taught and long since forgotten (for example "which shapes have diagonals which cross at right angles?" or perhaps "what is the sum of the exterior angles of an irregular octagon?")

Of course, TMUA doesn't directly test "which shapes have diagonals which cross at right angles?" (it would be nice it if did!). TMUA assumes these facts are known and uses them within other much trickier to answer questions.

So in Session #1 (continued in Session #2) we cover prerequisite maths required for TMUA including

• Fractions, decimals, percentages, ratios
• Standard form, non-decimal (binary, hexadecimal)
• Primes, prime factors, LCM and HCF
• Divisibility and divisibility rules
• Algebra - simplifying, factorising, expanding, roots of equations etc
• Remainder and factor theorem
• Simultaneous Equations
• Inequalities (graphical and algebraic)
• Functions - domain, range, composite
• Surds, power rules and basic logarithms
• Numerical approximations and rounding
• Mental maths techniques (there is no calculator allowed in the TMUA) 

Session #2: TMUA Prerequisite Maths Two

Session Length: 60 minutes. Continuing from Session #1 above, in Session #2 the prerequisite maths for TMUA that we cover includes 

• Straight lines, graphs and coordinate geometry
• Basic geometry and shape must-know facts (triangles, similarity and congruency, circles and circle theorems, parallelograms, polygons etc)
• Basic Trigonometry (trig values of common angles, periodicity, sine and cosine rules, trig formulae)
• Sets and Venn Diagrams
• Basic statistics (mean, mode, median, range etc) 
• Basic probability, permutations, combinations and counting 
• Must-know TMUA formulae (there is no formula booklet allowed in the TMUA)

Session #3: TMUA Exponents and Logarithms

Session Length: 60 minutes. A good understanding of how to manipulate exponents and logarithms, switch between the two forms, and solve exponential and logarithmic equations using the various log and power rules, as well as being able to draw sketches of log and exponent graphs (including their range, domain, shape and principal points), is absolutely essential for TMUA success. In this session, we dissect and analyse numerous past TMUA exam questions on exponents and logarithms, giving students focused practice in applying these techniques under exam conditions.

The TMUA sub-topics that we cover in this session are as follows:

1. Standard Exponent and Log TMUA Questions
2. Exponent and Log Graphs 
3. Exponent and Log Quadratics
4. Log Simultaneous Equations
5. Exponent Simultaneous Equations
6. Tricky Log Domains 

Session #4: TMUA Transformations

Session Length: 60 minutes. TMUA tests frequently on transformations - not only reflections, rotations, stretches and translations, but also more abstract transformations on general, often loosely defined, functions and how certain transformations might affect various characteristics such as its integral, derivative or turning points. By dissecting and analysing TMUA past paper questions on transformations, students should gain much confidence in dealing with any that appear on this years test.

The TMUA sub-topics that we cover in this session are as follows:

1. Translations, Reflections and Stretches
2. Rotations
3. Roots and Transformations
4. Graphs and Transformations
5. Integrals, Areas and Transformations

Session #5: TMUA Differentiation and Applications

Session Length: 60 minutes. Differentiation is a mainstay of the TMUA exam. It tests not only straightforward differentiation — mainly of polynomials or fractions involving various powers of 'x' — but also applications such as finding maxima and minima, gradients, second derivatives, turning points, and understanding the relationship between a function and its various derivatives. By dissecting and analysing a variety of past TMUA questions on differentiation and its applications, students build confidence in recognising which techniques to apply and how to solve problems efficiently under TMUA exam conditions.

The TMUA sub-topics that we cover in this session are as follows:

1. Standard Differentiation
2. Gradients and Normals
3. Turning Points
4. Increasing/Decreasing Functions
5. General Maximum/Minimum 
6. Maximum/Minimum of Objects

Session #6: TMUA Integration and Applications

Session Length: 60 minutes. Integration is a core topic tested in TMUA, often in ways that go far beyond straightforward direct or indirect integration of well-known functions (though these are tested too, primarily fairly straightforward polynomials). In this session, we tackle past TMUA integration and area questions, many of which require a deeper understanding of how integrals, and the areas they represent, behave when different conditions or transformations are applied to more generally defined functions.

The TMUA sub-topics that we cover in this session are as follows:

1. Standard Integrations
2. Areas under Curves
3. Differential Equations
4. Trapezium Rule
5. Integrating General Functions f(x)
6. Areas under General Functions f(x)

Session #7: TMUA Trigonometry

Session Length: 60 minutes. Trigonometry is a very frequent theme in TMUA. While some questions test 'standard' skills directly such as trig values of common angles, trig identities (especially cos^2(x) + sin^2(x)=1), the sine and cosine rules, and basic trigonometric graphs and ranges, most are more devious, involving trigonometry interspersed within other equations, graphs, or transformations. In this session, we work through past TMUA trigonometric linked questions and examine the tips, tricks and methods students should be familiar with in order to succeed.

The TMUA sub-topics that we cover in this session are as follows:

1. Standard Trig Equations and Inequalities
2. Exotic Trig Equations
3. Trig Graphs
4. Graphical Solutions to Equations involving Trig
5. Trig Maximum/Minimum
6. Trig Series and Sequences

Session #8: TMUA Quadratics (inc Hidden)

Session Length: 60 minutes. Quadratics and quadratic equations are a major feature of TMUA. While core skills like factorising, the quadratic formula (especially the discriminant and its implications), completing the square (very useful in finding maxima and minima), and graphing quadratics are essential, most TMUA questions go far beyond this. More often, TMUA hides quadratics within other question types, such as inequalities, trigonometry, logarithms, exponential equations, and graphical solutions. It also presents ‘unusual’ quadratic forms that aren’t immediately obvious. In this session, we tackle past TMUA questions involving quadratics, practising how to spot, solve, and interpret both standard and hidden forms efficiently and effectively under exam conditions.

The TMUA sub-topics that we cover in this session are as follows:

1. Standard Quadratics
2. Hidden Quadratics
3. Quadratic Inequalities
4. Transforming Quadratics
5. Quadratic Simultaneous Equations
6. Miscellaneous Quadratics

Session #9: TMUA Cubics and other Polynomials

Session Length: 60 minutes. TMUA loves cubics, and (to a lesser extent) higher-order polynomials. Core skills such as factorising, identifying roots and turning points, and sketching cubics are essential — particularly for recognising when a cubic (or higher-order polynomial) has a certain number of distinct or repeated roots in different regions of the graph. In this session, we work through past TMUA questions on cubics and higher-order polynomials to solidify understanding and build confidence in tackling these types of questions on the exam.

The TMUA sub-topics that we cover in this session are as follows:

1. Standard Cubics and other Polynomials
2. Cubic and Polynomial Inequalities
3. Roots of Cubics
4. Roots of Polynomials
5. Polynomial Calculus Linked

Session #10: TMUA Functions (inc Modulus and Symmetry)

Session Length: 60 minutes. In this session, we tackle past TMUA exam questions involving functions. The questions often go beyond simple evaluation or composition (though these are tested too, particularly on more exotic functions) and explore how functions — including modulus functions, even and odd functions, and other symmetrical or periodic functions — behave under various conditions and modifications such as transformations. By analysing and discussing these questions, students should gain confidence in recognising different function types and thus approach them effectively under TMUA exam conditions.

The TMUA sub-topics that we cover in this session are as follows:

1. Symmetry and Functions
2. Modulus and Functions
3. Recurrence and Pattern Functions
4. Functions and Calculus
5. Miscellaneous Functions

Session #11: TMUA Graphs and Graphical Solutions

Session Length: 60 minutes. TMUA examiners love graphs and graphical solutions, and this is an area where many students struggle in terms of question familiarity. While being able to sketch and interpret familiar graphs such as straight lines (which appear surprisingly frequently), quadratics, cubics, exponentials, logarithms and trigonometric functions is essential, most TMUA questions use this only as a starting point. The exam frequently tests students’ ability to solve 'algebraically challenging' equations and inequalities graphically, as well as to interpret intersections, transformations, and the impact of parameters on graphs. In this session, we cover a wide range of past TMUA questions to build fluency and confidence with these graphical techniques.

The TMUA sub-topics that we cover in this session are as follows:

1. Straight Lines and Graphs
2. Solving Equations Graphically
3. Inequalities and Graphs
4. Identifying Graphs

Session #12: TMUA Geometry and Shapes

Session Length: 60 minutes. Geometry and shapes feature regularly in TMUA, and many students initially struggle because these areas may not have been studied recently. A solid understanding of the characteristics of shapes such as triangles (including congruency and similarity), parallelograms, circles (including the equation of a circle), quadrilaterals, and polygons is essential, but most questions go far beyond these basic facts. TMUA frequently embeds geometric knowledge within other problem types, such as algebra, trigonometry, or logical reasoning. In this session, we tackle past TMUA geometry and shape questions to build confidence in recognising, analysing, and solving them efficiently under exam conditions.

The TMUA sub-topics that we cover in this session are as follows:

1. Circles and Circle Equations
2. Areas/Volumes of Shapes
3. Triangles and Congruency
4. Miscellaneous Geometry

Session #13: TMUA Statistics and Probability

Session Length: 60 minutes. Statistics and probability are not as commonly tested in TMUA as they are in other College Entrance Exams, however they do come up in most years' exam papers. Basic skills such as calculating mean, median, mode, range and simple probabilities, combined with basic counting and combinatorics, are essential, and questions then advance from these fundamentals to ask a related and usually less familiar question, often mixed with logic, inequalities and/or ordering. In this session, we tackle past TMUA questions on statistics and probability to ensure available marks in this topic are not easily squandered.

The TMUA sub-topics that we cover in this session are as follows:

1. Statistical Measures
2. Probability
3. Counting

Session #14: TMUA Arithmetic and Geometric Progressions

Session Length: 60 minutes. Arithmetic and geometric progression questions appear very regularly in TMUA. While knowing the formulae for nth terms and sums etc is essential, most questions go beyond these straightforward applications, either giving a series of 'clues' leaving the student to calculate missing information, or embedding arithmetic and geometric progressions within algebraic, functional, graphical or other seemingly unrelated contexts. In this session, we tackle past TMUA questions on arithmetic and geometric progressions to ensure students are prepared for the almost inevitable question on this topic that will come up.

The TMUA sub-topics that we cover in this session are as follows:

1. Arithmetic Progressions
2. Geometric Progressions
3. Combinations of Arithmetic and Geometric Progressions
4. Hidden Arithmetic and Geometric Progressions

Session #15: TMUA Binomial Expansions, Series and Sequences

Session Length: 60 minutes. Binomial expansions are a recurring theme in TMUA, and it is essential to know the Binomial Theorem and how to use it to find various coefficients and powers. Apart from arithmetic and geometric series (which we cover in Session #14), TMUA also likes questions on other sequences and series, usually deliberately designed to be unfamiliar to most students. Disentangling the sequence or series in order to then answer the question is often no mean task, and practice makes perfect, hence in this session we dissect a variety of TMUA questions on both binomial expansions and sequences and series (the latter part of this session links nicely with Session #16 on pattern finding).

The TMUA sub-topics that we cover in this session are as follows:

1. Standard Binomial Expansions
2. Complex Looking Binomial Expansions
3. Sequences (non arithmetic or geometric)
4. Series (non arithmetic or geometric)

Session #16: TMUA Pattern Finding and Word Problems

Session Length: 60 minutes. Pattern finding and word problems appear regularly in TMUA. While some questions may seem straightforward at first glance, there is usually some sort of twist rendering the eventual solution more elusive. Frequently, pattern finding is linked to disentangling bizarrely defined sequences or series (per Session #15) and equally frequently it is linked to a word problem or puzzle, which could literally be on any topic under the sun (to name but a couple, finding the number of attempts needed to guess a password and the various ways of arranging truth tellers and liars in a 3x3 grid). In this session, we tackle past TMUA questions to practise recognising and finding patterns, interpreting word problems and puzzles, and applying the right techniques efficiently to solve them under TMUA exam conditions.

The TMUA sub-topics that we cover in this session are as follows:

1. Pattern Finding
2. Word Problems

Session #17: TMUA Proofs

Session Length: 60 minutes. TMUA tests logic and reasoning a lot - indeed around half of Paper Two in one way or another is a logic or reasoning question of some form. Hence the final four sessions of our TMUA coaching course focus on this broad topic - certainly this is an area where students tend to need the most guidance (in our experience). In this session we focus on TMUA 'proof' questions, which are different from 'traditional' proofs in the sense that TMUA tends to provide a 'completed proof' and then asks on which line (if any) the proof becomes erroneous. There are certain 'errors' which tend to recur, and in this session we tackle a variety of past TMUA paper 'proof' questions so students should be ready when they see this question type on the exam.

The TMUA sub-topics that we cover in this session are as follows:

1. Algebraic Proofs
2. Verbose Proofs
3. Logical Flow of Proofs
4. Proofs and False/Missing Solutions

Session #18: TMUA Necessary/Sufficient Questions

Session Length: 60 minutes. In this second of four sessions on the extremely popular TMUA topic of logic and reasoning, we examine and dissect 'necessary/sufficient' questions, which are among the most unfamiliar to most students taking the exam. The basic premise of X being 'necessary' or 'sufficient' for Y is quite easy to grasp, but many questions twist the wording and frequently invoke 'easy to misinterpret' or 'obviously correct but actually wrong' options. There is no doubt that practice makes perfect, and in this session we tackle a good number of  'necessary/sufficient' TMUA questions.

The TMUA sub-topics that we cover in this session are as follows:

1. 'Necessary' Questions
2. 'Sufficient' Questions
3. 'Equivalent' Questions

Session #19: TMUA True/False; If/Only If; and Contrapositive/Converse

Session Length: 60 minutes. In this third of four sessions on the extremely popular TMUA topic of logic and reasoning, we examine 'True or False' TMUA questions (which occasionally need identification of contrapositive and/or converse to answer correctly) and 'If; Only If; and If and Only If' TMUA questions. These questions tend to be unfamiliar to TMUA students, especially the 'which of the following MUST be true' type questions and 'A only if B' or 'A if B' which get muddled up. By analysing a variety of past TMUA questions on this topic, students should be able to avoid the pitfalls that TMUA likes to present.

The TMUA sub-topics that we cover in this session are as follows:

1. True/False General
2. True/False Inequalities
3. If, Only If and Iff
4. Contrapositive and Converse

Session #20: TMUA Counterexamples, Negations and Other Logic

Session Length: 60 minutes. In this fourth of four sessions on the extremely popular TMUA topic of logic and reasoning, and the final session of our course, we examine 'counterexample' questions (where a statement is given together with a list of possible counterexamples - or vice versa); 'negation' questions (where a wordy and usually complex mathematical explanation of a theorem or idea needs to be negated); and 'other' logic and reasoning questions which TMUA occasionally likes to throw into the mix. Familiarity with these questions is the key, and in this session we analyse a variety of past TMUA questions to ensure that is the case, maximising students' chances of TMUA success.

The TMUA sub-topics that we cover in this session are as follows:

1. Standard Counterexamples
2. Further Counterexamples
3. Negations
4. Other Logic