Bagrut Mathematics (3/4/5 Units) common mistakes
Bagrut Mathematics (3/4/5 Units) · Understand the traps and how to avoid them.
When solving an equation with a parameter or after multiplying by an expression containing x, the candidate divides both sides by that expression (or by x itself) and silently loses the solution x = 0 or the parameter value that makes the divisor zero, reporting fewer roots than the equation actually has.
School algebra drills 'cancel common factors' as a simplification reflex, so division feels like progress rather than a lossy operation. Candidates do not track that division is only valid when the divisor is nonzero, and they rarely check their answer count against the polynomial degree.
What to do instead
Before dividing, write the condition 'divisor ≠ 0' explicitly on the page. Prefer factoring and using the zero-product principle: move everything to one side, factor, and set each factor to zero, which keeps the x = 0 branch. After solving, compare the number of roots with the degree of the original polynomial and investigate any shortfall.
When solving an inequality, the candidate multiplies or divides by a negative number (or by an expression whose sign depends on x) without reversing the inequality sign, and writes the solution interval as if it were an equation.
Equation-solving habits are overlearned, and the sign-reversal rule is a one-line rule that is easy to forget under time pressure. Multiplying by a variable expression is worse: the candidate treats it as a positive constant when its sign is unknown.
What to do instead
Never multiply an inequality by an expression containing x. Instead move all terms to one side, combine over a common denominator, factor the numerator and denominator, and build a sign table over the critical points, testing one value in each interval. When a numeric negative multiplier is unavoidable, immediately write the reversed sign as the next line.
When sketching or analysing a function, the candidate applies transformations in the wrong order or direction, for example reading y = f(2x − 6) as a shift of 6 right then a stretch, or moving y = f(x + 3) three units right instead of left.
Horizontal transformations act 'backwards' relative to intuition because they operate on the input before f is applied. Candidates memorize 'plus means right' from number-line habits and do not rewrite the expression in factored form.
What to do instead
Always rewrite the inside as a factor: f(2x − 6) = f(2(x − 3)), so the stretch by 1/2 comes first and the shift is 3 right. Verify with three concrete points, computing the image of x = 0, 1, 2 under both the original and the transformed rule, and confirm the plotted points match.
The candidate treats exponent rules as if they applied to every operation, writing (a + b)^2 = a^2 + b^2, √(a + b) = √a + √b, or log(x + y) = log x + log y, and builds a whole solution on that false identity.
The distributive law is the dominant algebraic pattern, so the brain extends it to squaring and to roots and logs. These errors survive because the false step often produces a plausible-looking answer that is never tested numerically.
What to do instead
Keep a short 'illegal moves' list and, on every multi-step simplification, substitute one easy number (x = 2, y = 3) into the line before and after the step. If the two lines give different values, the step is invalid. For logs, state the product, quotient and power laws explicitly before using them.
In exponential and logarithmic problems the candidate solves for x but ignores the domain: taking log of a possibly negative or zero expression, accepting a root that makes an original log argument non-positive, or forgetting that a^x is always positive so a^x = −5 has no solution.
Domain restrictions are taught as a separate topic and are not integrated into the solving routine. Once the algebra produces a number, the candidate stops, treating the domain check as optional decoration.
What to do instead
At the very first line of a log equation, write the required conditions (each argument > 0, base > 0 and ≠ 1) and keep them visible. After finding candidates, test each against those conditions and cross out rejected ones with a written reason. For exponentials, state the range a^x > 0 before claiming no solution.
In trigonometry the candidate finds one solution to sin x = k or cos x = k and stops, missing the second solution in [0, 360°] and all solutions obtained by adding 360°k, or applies the sine rule in a triangle where the given data produce the ambiguous case.
The inverse-trig function on a calculator returns a single principal value, which creates the false impression that the equation has one answer. The symmetry of the unit circle is known in theory but not used as a solving tool.
What to do instead
Solve every trig equation by sketching the unit circle or the graph over one full period and marking all intersections. Use the reference angle plus the symmetry rules (180° − θ for sine, 360° − θ for cosine) and finish with the general solution plus 360°k or 180°k. For the sine rule, always check whether the supplementary angle is also compatible with the angle sum.
In a sequence or series problem the candidate confuses the n-th term with the sum of the first n terms, substitutes n instead of n − 1 into the term formula, or uses a finite geometric sum formula without checking whether the ratio equals 1.
The notation a_n and S_n look interchangeable, and the arithmetic sequence formula a_n = a_1 + (n − 1)d is memorized without the reasoning that there are n − 1 steps from the first term. The special case r = 1 divides by zero but is rarely flagged.
What to do instead
Write the definitions side by side at the top of every solution: a_n is one term, S_n is a total, and S_n − S_{n−1} = a_n. Test any formula on a tiny case (n = 1, 2, 3) before applying it to the actual problem, and state 'if r = 1 then S_n = n·a_1' before using the general geometric sum.
In calculus the candidate differentiates or integrates without regard to the variable of differentiation, drops the inner derivative in a chain-rule or substitution step, and omits the constant of integration or the limits when the problem asks for a definite integral.
Procedural fluency outruns structural understanding: the rules are applied as pattern matching on the outer function, and the '+ C' feels cosmetic rather than mathematical. Substitution is often done halfway, changing the integrand but not the differential.
What to do instead
Before differentiating, name the inner function u and write du/dx explicitly; before substituting in an integral, write u, du and the transformed limits in a small box, then convert everything, including dx and the bounds. End every indefinite integral with '+ C' and every area or definite-integral answer with an evaluation of the antiderivative at both limits.
In Euclidean geometry the candidate writes a proof that asserts the desired conclusion, using the property to be proved as a reason, or cites a theorem by name without stating the specific given conditions (for example claiming triangles are congruent from SSA, or using 'alternate angles are equal' without parallel lines).
Proof is the least practised skill and the candidate reasons backwards from the diagram, which already looks correct. Diagram-driven reasoning also ignores that the figure may not be drawn to scale or may show only one configuration.
What to do instead
Build the proof as a two-column chain in which every line is either given, a construction, or a named theorem with its hypotheses checked against the previous lines. Never cite a fact that appears later in the chain. Redraw the figure from the written givens alone, without the conclusion, to confirm the reasoning does not depend on the picture.
In analytic geometry and in probability the candidate substitutes into a formula without checking its preconditions: using the perpendicular-gradient rule without confirming the lines are not vertical, computing distance or midpoint with coordinates subtracted in inconsistent order, or using P(A and B) = P(A)P(B) for events that are not independent and P(A or B) = P(A) + P(B) for events that are not disjoint.
Formulas are recalled as isolated strings, and the conditions attached to them are learned separately and then forgotten. In probability the word 'or' and the phrase 'and' invite simple addition and multiplication without an independence check.
What to do instead
For each formula, write the precondition next to it before substituting: slope rule needs finite slopes; independence needs P(A|B) = P(A) or a stated random independent mechanism; disjointness needs an empty intersection. Draw a Venn diagram or two-way table first and read the required probability off it, then confirm the formula agrees.