Estonian Basic-School Mathematics Exam cheat sheet

Order of operations: brackets first, then powers and roots, then multiplication and division left to right, then addition and subtraction left to right.

Sign rules: (+)(+)=+, (−)(−)=+, (+)(−)=−; for division the same rules apply. Subtracting a negative equals adding: a−(−b)=a+b.

Divisibility: by 3 or 9 if the digit sum is divisible by 3 or 9; by 4 if the last two digits form a multiple of 4; by 5 if the last digit is 0 or 5.

A rational number can be written as a/b with integers a, b and b≠0. Its decimal expansion is terminating or eventually repeating; irrational numbers (√2, π) are neither.

Absolute value |a| is the distance of a from 0 on the number line, so |a|≥0 and |−a|=|a|.

Powers: a^m·a^n=a^(m+n), a^m/a^n=a^(m−n), (a^m)^n=a^(mn), a^0=1 (a≠0), a^(−n)=1/a^n.

Estimation: round each number to one significant figure to check magnitude; if the exact answer and the estimate differ by a factor of 10, suspect a decimal or place-value error.

Standard form a·10^n with 1≤a<10; multiplying shifts the exponent up, dividing shifts it down.

Expand: a(b+c)=ab+ac; (a+b)(c+d)=ac+ad+bc+bd; (a+b)²=a²+2ab+b²; (a−b)²=a²−2ab+b²; (a+b)(a−b)=a²−b².

Factorise by taking out the common factor first, then recognise a difference of squares a²−b²=(a+b)(a−b) or a perfect square trinomial.

Solving ax+b=c: subtract b from both sides, then divide by a (a≠0). Whatever operation is applied to one side must be applied to the other.

Solving a linear inequality: the direction of the inequality reverses only when multiplying or dividing both sides by a negative number.

A linear equation in one unknown has one solution unless the variable cancels and a false statement remains (no solution) or a true statement remains (infinitely many).

System of two linear equations: substitution (express one variable from one equation) or elimination (add/subtract multiples of equations to remove a variable).

Rearranging a formula for a chosen variable uses the same inverse-operation sequence as solving an equation, treating all other letters as constants.

Quadratic equation ax²+bx+c=0: x=(−b±√(b²−4ac))/(2a); the discriminant D=b²−4ac gives two, one, or no real roots as D>0, D=0, D<0.

A function assigns exactly one output to each input; the graph passes the vertical-line test.

Linear function y=kx+b: k is the slope (change in y per unit change in x), b is the y-intercept. Slope from two points: k=(y₂−y₁)/(x₂−x₁).

Parallel lines have equal slopes; perpendicular lines have slopes whose product is −1.

Direct proportionality y=kx passes through the origin; inverse proportionality y=k/x is a hyperbola with the axes as asymptotes, and k≠0.

Quadratic function y=ax²+bx+c has a parabola graph: opens upward if a>0, downward if a<0; the vertex x-coordinate is −b/(2a) and it is the axis of symmetry.

To read a graph: find the value of y for a given x by moving vertically to the curve then horizontally to the y-axis; solve f(x)=c by finding where the horizontal line y=c meets the graph.

Zero of a function is the x-value where f(x)=0, i.e. where the graph crosses the x-axis.

Rate of change over an interval equals the slope of the straight line joining the two graph points.

Triangle angle sum is 180°; exterior angle equals the sum of the two non-adjacent interior angles. Quadrilateral angle sum is 360°.

Pythagoras: in a right triangle a²+b²=c², where c is the hypotenuse. Converse: if a²+b²=c², the triangle is right-angled.

Area formulas: triangle A=½ah; parallelogram A=ah; trapezium A=½(a+b)h; circle A=πr², circumference C=2πr.

Volume: prism/cylinder V=Ah (base area times height); pyramid/cone V=⅓Ah; sphere V=(4/3)πr³. Total surface area is the sum of all face areas.

Similar figures: corresponding angles equal and corresponding lengths in the same ratio k. Areas scale by k², volumes by k³.

Trigonometry in a right triangle: sin α=opposite/hypotenuse, cos α=adjacent/hypotenuse, tan α=opposite/adjacent.

Angle sum of a polygon with n sides is (n−2)·180°; each interior angle of a regular n-gon is (n−2)·180°/n.

Unit conversion: 1 m=100 cm, 1 m²=10 000 cm², 1 m³=1 000 000 cm³, 1 L=1 dm³=1000 cm³, 1 ha=10 000 m².

Arithmetic mean = sum of values / number of values; it is affected by extreme values, the median is not.

Median: order the data and take the middle value, or the mean of the two middle values when n is even. Mode is the most frequent value (may be absent or multiple).

Range = maximum − minimum. It measures spread but ignores all values between the extremes.

In a bar chart the height of each bar gives the frequency; in a histogram, frequency equals bar height times bar width when widths differ.

Pie chart: the angle of a sector equals (frequency/total)·360°; percentages convert to angles by multiplying by 3.6.

Probability of an event P(A) = number of favourable outcomes / number of equally likely outcomes, so 0≤P(A)≤1.

P(A does not occur)=1−P(A). For mutually exclusive events P(A or B)=P(A)+P(B).

Relative frequency from an experiment approaches the theoretical probability as the number of trials grows.

Translate: 'sum' → +, 'difference' → −, 'product' → ×, 'quotient' → ÷, 'is/was' → =, 'per' or 'each' → rate multiplier.

Define the unknown with a letter in words first (e.g. let x be the price of one notebook), then build the equation from the stated relations.

Distance–speed–time: s=v·t, so v=s/t and t=s/v. For average speed use total distance divided by total time, not the mean of the speeds.

Percentage change = (new − old)/old · 100%; a price increased by p% is multiplied by (1+p/100), a decrease by (1−p/100).

Successive percentage changes multiply: +20% then −20% gives 1.2·0.8=0.96, i.e. a 4% net decrease.

Mixture and work problems: set up a table of amount × concentration or rate × time, and require the parts to sum to the total.

After solving, check the answer against the original wording and reject values that are physically impossible (negative length, non-integer number of people).

A proof or justification must start from given data and known facts and reach the claim by stated steps; a single verified example does not prove a general statement.

A counterexample disproves a general claim; one is enough. Check statements about all numbers by testing 0, 1, negatives and fractions.

Write units in every intermediate result and in the final answer; a numeric answer without units is incomplete in applied tasks.

Rounding rule: round only the final answer unless instructed otherwise; state the accuracy used (e.g. 2 decimal places, 3 significant figures).

Use ≈ for rounded values and = only for exact equality. Keep the same unit within a calculation; convert before computing.

Answer the question that was asked: if it asks for a quantity, give the number with unit and a short sentence of interpretation.

Geometry justification: cite the reason for each step (angle sum of triangle, corresponding angles in parallel lines, Pythagoras, similarity ratio).

When comparing results, state the criterion explicitly (larger value, smaller cost per unit, higher percentage) and quote both figures.

Addition and subtraction of fractions: common denominator, then add or subtract numerators; multiplication is numerator×numerator over denominator×denominator.

Division by a fraction equals multiplication by its reciprocal: a÷(b/c)=a·(c/b).

Percent means per hundred: p% = p/100. Finding p% of a quantity means multiplying by p/100.

Comparing numbers: convert fractions, decimals and percentages to the same form, e.g. 3/8=0.375=37.5%.

A negative number raised to an even power is positive, to an odd power is negative: (−2)⁴=16, (−2)³=−8.

Square roots: √(a·b)=√a·√b for a,b≥0; √(a²)=|a|. The square root of a negative number is not real.

Ordering on the number line: for a<b, adding the same number keeps the order; multiplying by a negative reverses it.