Free TMUA guide
TMUA trap families from question patterns
The same mathematical mistake can appear in algebra, geometry, number or reasoning. These families are distilled from the kinds of patterns that appear across official preparation materials and the specification. The examples here are original and do not quote a live question.
Algebra traps: a valid move needs a condition
| Trap family | What goes wrong | Block it with |
|---|---|---|
| Unrestricted division | A factor can be zero, so cancellation loses a case | Write the non-zero condition beside the division |
| Squaring an equation | Squaring can make a false equality look true | Substitute every candidate into the original equation |
| Sign drift | An inequality changes direction or a negative factor is ignored | Mark the sign before multiplying or dividing |
| Option-shaped algebra | The route is bent to reach a familiar-looking option | Keep the equation separate from the options until the end |
Worked example Check the original equation
Solving gives two real candidates, not one.
If the original stem also says , only remains. The domain is part of the answer, not decoration.
Why this works: Keep every branch until the original conditions remove it.
Why the tempting route fails: Writing only the positive root because it is the familiar principal root loses a valid algebraic branch before the stem has been used.
Geometry and number traps: scale and structure matter
- Linear scale factors apply once to lengths, twice to areas and three times to volumes.
- A diagram that looks symmetric does not prove symmetry. Mark the equalities you are allowed to use.
- A pattern in the first few integers is not a divisibility proof. Track parity, factors or remainders.
- An endpoint can behave differently from the interior of an interval. Test it separately when the inequality permits it.
Worked example Area needs the scale factor twice
A smaller shape is similar to a larger shape with linear scale factor . The larger area is .
The smaller area is . Applying once would give a length relationship, not an area relationship.
Why this works: Name the dimension of the requested quantity before applying the scale factor.
Why the tempting route fails: Using the length factor directly on area is a unit mismatch even when the final number looks tidy.
Reasoning traps: direction, quantifiers and cases
| Trap family | Typical wrong turn | Fast audit |
|---|---|---|
| Converse error | Treating as | Find a case where holds and fails |
| Universal overreach | Several examples are treated as proof for all values | Search the smallest edge, odd, negative or zero case in the domain |
| Existential swap | A value that works is treated as every value | Write 'there exists' or 'for all' in your own words |
| Incomplete cases | Even and odd cases are not both covered | List the cases before simplifying |
| First-error drift | The final wrong line is blamed instead of the first unsupported move | Check each arrow from one line to the next |
Worked example A picture is not a counterexample
A statement claims that a relation holds for every triangle with a stated angle condition. A hand-drawn triangle that does not satisfy the angle condition cannot disprove it.
Build a symbolic or numerical case inside the domain. If the condition is not met, the drawing tests a different claim.
Why this works: A counterexample must satisfy every hypothesis and fail the conclusion.
Why the tempting route fails: A visually suggestive sketch outside the hypotheses is evidence about a different problem.
Reading traps: the question can change the task
- Circle words such as not, except, least, greatest, always and could.
- Copy the requested object: value, number of solutions, statement, area or proof step.
- Check whether an option gives a condition, a consequence or an example.
- At the end, read the stem once more without looking at your chosen option.
Sources
UAT-UK TMUA preparation materials. Official preparation papers and worked answers for comparing your own trap log with published question patterns. Open the materialshttps://esat-tmua.ac.uk/tmua-preparation-materials/
TMUA Content Specification. The current mathematical and reasoning boundaries behind the families in this guide. Read the specificationhttps://uat-wp.s3.eu-west-2.amazonaws.com/wp-content/uploads/2024/05/03165619/TMUA_Content_Specification.pdf
The trap families describe study patterns, not a guarantee that any particular question will contain a particular distractor.