ILTS Mathematics (208) common mistakes
ILTS Mathematics (208) · Understand the traps and how to avoid them.
When simplifying or combining rational expressions, the candidate multiplies both sides of an equation by the least common denominator instead of applying the LCD only to the terms being added, producing an expression that is no longer equivalent to the original.
Procedural memory from solving equations overrides the distinction between an equation (where multiplying both sides preserves solutions) and an expression (where it changes value). Under time pressure the candidate grabs the most recently practiced move.
What to do instead
Before touching the problem, state aloud whether the task is 'solve' or 'simplify.' For simplify tasks, write the LCD above each fraction and rewrite each numerator with a bracket, then combine only numerators. Check with a test value such as x = 2 in both the original and final expression; if the values differ, the operation was invalid.
The candidate computes a probability of a compound event by adding probabilities of overlapping events, or by multiplying probabilities of events that are not independent, without checking the overlap or the sampling condition.
The 'or means add, and means multiply' shortcut is learned early and survives into adulthood. The candidate does not habitually ask whether events are mutually exclusive or whether sampling is with or without replacement.
What to do instead
Adopt a fixed two-question protocol for every compound event: (1) Do the events overlap? If yes, subtract the intersection. (2) Does the first event change the probability of the second? If yes, condition. Then verify with a small sample space or a two-way table. Build the habit of drawing the sample space for small problems before computing.
On proportional-reasoning items, the candidate sets up a ratio with quantities in inconsistent order (miles per hour written as hours per mile) or uses an additive comparison where a multiplicative one is required.
Ratios are often taught as a formula to memorize rather than as a comparison of two quantities with units. The candidate does not track units through the setup.
What to do instead
Write the ratio as a labeled fraction with units attached to each number, and cancel units before computing. For comparison problems, ask explicitly 'how many times as many' versus 'how many more' and choose the operation accordingly. Practice by writing the unit rate as a sentence with 'per' in it.
The candidate solves a multi-step word problem correctly but answers a different quantity than the one requested, leaving a partial result (for example, the cost of one item rather than the total, or the discount rather than the final price).
Working memory is consumed by the computation, so the goal state is not re-checked. Textbook problems often end at the intermediate step, reinforcing the habit.
What to do instead
Before computing, underline the question and write a one-line 'answer will be' statement with units. After computing, read the underlined question again and compare. On constructed-response items, add a final sentence that restates the answer with units, since scorers award credit for the correct quantity.
The candidate treats a transformation of a function as if it acts in the intuitive direction, shifting f(x) left for f(x - 3) or misordering horizontal stretches with horizontal shifts.
Horizontal transformations operate on the input, so they run opposite to the sign shown. The candidate memorizes rules without connecting them to a table of values.
What to do instead
For every transformation, build a three-row table: x, the inner expression, and the output. Track where the key points (vertex, intercepts) move. State the rule as 'inside the function changes the input, so it acts backwards.' Verify with a point test.
The candidate confuses the conditions for congruence and similarity, applying SSS or SAS to prove similarity, or assuming triangles are similar because two sides are proportional without checking the included angle.
The criteria are memorized as a list of letter combinations without the underlying logic of rigid motion versus dilation plus rigid motion.
What to do instead
Anchor each criterion to a transformation: congruence means a rigid motion maps one figure onto the other; similarity means a dilation followed by a rigid motion. For each criterion, write the included or corresponding parts explicitly. Reject any criterion that does not guarantee the mapping.
When computing measures of center and spread, the candidate reports the mean and standard deviation for skewed data with outliers, or fails to notice that adding a constant to every data value shifts the mean but not the standard deviation.
Mean and standard deviation are the default outputs of calculators and software, so the candidate does not evaluate whether they are appropriate summaries. The transformation rules for summary statistics are rarely practiced.
What to do instead
Before choosing a summary, inspect a dot plot or five-number summary and ask about skew and outliers; if present, report median and IQR with a reason. Memorize the shift/scale rules by deriving them on a small data set of five numbers, then re-derive rather than recall on the exam.
The candidate models a real context with a linear function when the rate of change is itself changing, or fits a line to data that clearly curve, without checking residuals or the context's mechanism.
Linear models are the most practiced, and the candidate does not connect the choice of model to the question of constant versus varying rate of change.
What to do instead
For each modeling item, state the rate of change in words and ask whether it is constant. If the context involves growth by a percentage, doubling, or acceleration, choose exponential or quadratic. Plot or reason about residuals and confirm the sign of the predicted trend matches the context.
The candidate uses an algorithm or procedure to justify a mathematical claim, offering 'I multiplied both sides' or 'I used the formula' instead of a mathematical argument grounded in definitions, properties, or counterexamples.
School mathematics rewards answer-getting, so justification is underdeveloped. The candidate does not distinguish showing work from proving.
What to do instead
Practice writing two-sentence justifications that name the property used (distributive property, definition of even integer, triangle inequality). For true/false items, try to construct a counterexample first; if one exists, the statement is false and the counterexample is the justification.
The candidate computes an exact answer and reports it with inappropriate precision, such as 12.347 calculators for a context that permits only whole items, or fails to state units and reasonableness in a modeling response.
The candidate treats the numeric output as the endpoint and does not revisit the context to interpret it. Rounding rules are applied mechanically rather than by context.
What to do instead
End every modeling problem with an interpretation sentence: quantity, unit, and whether it is reasonable. Ask 'can this be fractional or negative in this context?' and round accordingly, stating the rounding decision. On constructed-response items, this sentence is often a scored element.