Praxis Math 5165 common mistakes
Praxis Math 5165 · Understand the traps and how to avoid them.
Simplifies a radical or a power of a sum as if the operation distributed: writes (a + b)² = a² + b², or computes a distance as √(3² + 4²) = 3 + 4 = 7, or reduces √(x²) to x.
Distributivity of multiplication over addition is learned so thoroughly that it gets over-generalized to exponents, radicals, and norms - the surface pattern (an operation applied to a sum) looks identical to the legitimate rules.
What to do instead
Keep a short written list of non-rules (√(a+b), (a+b)², (a+b)⁻¹, log(a+b)) and test every simplification with a = 3, b = 4 before finalizing. Remember √(x²) = |x|, so state the absolute value when the sign is unknown.
Multiplies radicals of negative numbers directly, writing √(−4)·√(−9) = √36 = 6, or drops a sign when using i² = −1 in a quotient such as (3 + 2i)/(1 − i).
The product rule √a·√b = √(ab) is remembered without its precondition that a and b be nonnegative, and the imaginary unit is treated as a decoration rather than a quantity with a defined square.
What to do instead
Convert every negative radicand to i√n before multiplying, and carry the √(−1) factors as a separate bookkeeping column until the last step. For division, multiply numerator and denominator by the conjugate and re-expand i² = −1 explicitly.
Squares both sides of a radical equation, or clears the denominator of a rational equation, then reports every algebraic solution without substituting back into the original.
The verification step feels like busy work under time pressure, and squaring feels reversible - so the answer is submitted as soon as the algebra terminates.
What to do instead
Before solving, write the restrictions in the margin: denominator ≠ 0, radicand ≥ 0, log argument > 0. After solving, substitute each candidate into the original equation in one line and discard the ones that fail.
Reads f(g(x)) as the product f(x)·g(x), or describes y = f(2x − 6) as a shift of 6 units right rather than 3.
Composition notation is visually adjacent to multiplication, and horizontal transformations are counterintuitive because the scaling acts before the shift when read left to right.
What to do instead
Factor the argument before transforming: f(2x − 6) = f(2(x − 3)) → right 3, compress by ½. For composition, evaluate inside-out with a concrete number such as f(g(2)) to confirm which function acts first.
Solves a triangle given two sides and a non-included angle (SSA) with the Law of Sines and reports only the acute angle, or reaches for the Law of Sines when the given data are two sides and the included angle (SAS).
Practice problems usually have one clean answer, so the second triangle produced by sin θ = sin(180° − θ) is never considered; and both laws are described as relating sides and angles, so the trigger conditions blur together.
What to do instead
Use a decision table: ASA, AAS, SSA → Law of Sines; SAS, SSS → Law of Cosines. For SSA, explicitly compute θ₂ = 180° − θ₁ and check whether θ₂ plus the given angle is under 180°; if so, present both triangles.
Leaves the calculator in degree mode while evaluating sin(π/6), or gives arcsin(0.5) = 150° as the answer instead of the principal value.
The mode indicator is easy to miss and persists between problems, and inverse trig ranges are memorized as 'the answer' rather than as a restriction on the principal value.
What to do instead
Write the mode (DEG or RAD) next to every trig computation before pressing keys, and decide from the input form (π present → radians). Memorize the principal ranges: arcsin in [−90°, 90°], arccos in [0°, 180°], arctan in (−90°, 90°).
Substitutes the slant height of a cone or pyramid for h in V = (1/3)πr²h, or scales the volume of a similar solid by the linear factor k instead of k³.
Formulas are stored as strings of symbols without the geometric meaning of each variable, and similarity is first learned with side lengths, so the length factor is the one that gets reused.
What to do instead
Sketch the solid and label which segment is perpendicular height versus slant height before substituting. Apply the scaling law explicitly: lengths scale by k, areas by k², volumes by k³.
Computes P(A and B) = P(A)·P(B) for two cards drawn without replacement, or answers with P(B|A) when the question asked for P(A|B).
The independence formula is the one drilled most often and becomes the default multiplication rule, while conditional notation is read left to right as 'A then B'.
What to do instead
Before multiplying, ask 'does the first outcome change the second?' If yes, adjust the second factor (e.g., 5/52 · 4/51). For conditionals, write the definition P(A|B) = P(A∩B)/P(B) and substitute numbers rather than reasoning from the order of the letters.
Uses σ instead of σ/√n for the standard deviation of a sample mean, or interprets a 95% confidence interval as 'a 95% chance that μ lies in this interval' and a p-value as the probability that the null hypothesis is true.
Population spread and sampling-distribution spread collapse into one mental category of 'variability', and interval and p-value procedures are learned as keystroke sequences rather than as probability statements.
What to do instead
Draw the three-level ladder - population (σ), one sample (s), sampling distribution (σ/√n) - and label the spread at each level. Use fixed sentence templates: for a CI, 'if we repeated the sampling many times, about 95% of intervals constructed this way would capture μ'; for a p-value, 'the probability of data at least this extreme, assuming H₀ is true'.
Antidifferentiates a linear composition by raising the power and dividing by the new power only - ∫(2x + 1)³ dx → (2x + 1)⁴/4 - omits + C on an indefinite integral, or gives f′(a) when the average rate of change over [a, b] was requested.
Differentiation rules are run forward as pattern matches, so the inner derivative factor 1/u′ silently vanishes; and 'rate of change' is treated as a single idea rather than two distinct quantities.
What to do instead
Use ∫(ax + b)ⁿ dx = (ax + b)ⁿ⁺¹/(a(n + 1)) and then differentiate the result mentally to confirm it returns the integrand. Attach + C to every indefinite integral. For rate questions, check units: f′(a) is output per input at a single point, while the average rate is (f(b) − f(a))/(b − a).