Bulgaria NVO Grade 7 Mathematics 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.

For fractions a/b + c/d, first rewrite with common denominator bd: (ad + bc)/(bd); for division multiply by the reciprocal: (a/b) ÷ (c/d) = (a/b)·(d/c).

To add or subtract decimals, align the decimal points and fill empty places with zeros; the decimal point in a product has as many places as the two factors combined.

Multiplying by a power of 10 shifts the decimal point: ×10^n moves it n places right, ÷10^n moves it n places left.

Rules for signs: (+)(+)=+, (−)(−)=+, (+)(−)=−; a negative number squared is positive, but −a² means −(a²) and is negative for a ≠ 0.

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

a^m · a^n = a^(m+n), a^m ÷ a^n = a^(m−n), (a^m)^n = a^(mn), a^0 = 1 for a ≠ 0, and a^(−n) = 1/a^n.

√(a·b) = √a·√b for a, b ≥ 0; √(a²) = |a|; a square root of a negative number is not defined in this course.

A number is divisible by 3 (or 9) exactly when the sum of its digits is; by 2 when the last digit is even; by 5 when it ends in 0 or 5.

A monomial is a product of numbers and variables with non-negative integer exponents; its degree is the sum of the variable exponents, and like monomials have identical variable parts.

Combine like terms by adding their coefficients: 5x²y − 8x²y = −3x²y; unlike terms such as 3x and 3x² cannot be merged.

Distributive law: a(b + c) = ab + ac and a(b − c) = ab − ac; when a minus sign precedes brackets, every sign inside changes.

(a + b)² = a² + 2ab + b²; (a − b)² = a² − 2ab + b²; (a + b)(a − b) = a² − b².

Difference of squares factors as a² − b² = (a − b)(a + b); a perfect square trinomial factors as a² ± 2ab + b² = (a ± b)².

Common-factor extraction is the first factoring step: ax + ay = a(x + y); check by multiplying back.

To multiply polynomials, multiply every term of one by every term of the other and then collect like terms.

x² + (p + q)x + pq = (x + p)(x + q): find two numbers whose product is the constant and whose sum is the x-coefficient.

Adding or subtracting the same number on both sides of an equation, or multiplying/dividing both sides by a nonzero number, keeps the solution set unchanged.

To solve a linear equation, expand brackets, clear fractions by multiplying by the common denominator, collect variable terms on one side and constants on the other.

A linear equation ax + b = 0 has exactly one solution when a ≠ 0, no solution when a = 0 and b ≠ 0, and every number as a solution when a = 0 and b = 0.

Multiplying or dividing an inequality by a negative number reverses the inequality sign; adding or subtracting a number never reverses it.

Solve a system by substitution (express one variable from one equation) or by elimination (make coefficients opposite and add the equations).

A system of two linear equations has one solution if the lines intersect, none if the lines are parallel, and infinitely many if the lines coincide.

For a compound inequality a < x + c ≤ b, apply the same operation to all three parts to isolate x.

Check a solution by substituting it back into the original equation or inequality; for inequalities, test whether the resulting statement is true.

A point in the plane is given by an ordered pair (x, y): x is the horizontal coordinate, y the vertical; the origin is (0, 0).

The linear function y = kx + m has a straight-line graph with slope k and y-intercept m; the line crosses the y-axis at (0, m).

Slope k = (y₂ − y₁)/(x₂ − x₁) measures change in y per unit change in x; k > 0 means increasing, k < 0 decreasing, k = 0 horizontal.

A proportional relationship y = kx passes through the origin; its graph is a line through (0, 0) and the ratio y/x is constant.

Two lines are parallel exactly when their slopes are equal (k₁ = k₂); they are perpendicular when k₁·k₂ = −1.

To find where a line meets the x-axis, set y = 0 and solve for x; to find the y-intercept, set x = 0.

The graph of y = kx + m is obtained from y = kx by shifting m units vertically: up if m > 0, down if m < 0.

A point lies on the graph of y = f(x) exactly when its coordinates satisfy the equation.

The angles of a triangle sum to 180°; an exterior angle equals the sum of the two non-adjacent interior angles.

A triangle is isosceles exactly when two sides are equal; then the angles opposite those sides are equal, and conversely.

Congruence tests for triangles: SSS, SAS, ASA (with AAS) and, for right triangles, hypotenuse–leg; AAA does not prove congruence.

Similar triangles have equal corresponding angles and proportional corresponding sides; the ratio of their areas equals the square of the similarity ratio.

The sum of the interior angles of an n-gon is (n − 2)·180°; the sum of exterior angles of any convex polygon is 360°.

A parallelogram has opposite sides equal and parallel, opposite angles equal, and diagonals that bisect each other.

A rhombus has all sides equal and perpendicular diagonals; a rectangle has equal diagonals; a square has both properties.

The midline of a triangle is parallel to the third side and half its length; the midline of a trapezoid is half the sum of the bases.

In a right triangle, the median to the hypotenuse equals half the hypotenuse, and the side opposite a 30° angle is half the hypotenuse.

Rectangle: perimeter P = 2(a + b), area S = ab; square: P = 4a, S = a².

Triangle: area S = (1/2)·a·h, where a is a base and h the height to that base; perimeter is the sum of the three sides.

Parallelogram: S = a·h; trapezoid: S = ((a + b)/2)·h, where a and b are the parallel bases.

Circle: circumference C = 2πr = πd, area S = πr²; a sector with central angle α has area (α/360)·πr² and arc length (α/360)·2πr.

Pythagorean theorem: in a right triangle a² + b² = c², where c is the hypotenuse; it gives the distance and diagonal lengths.

For composite figures, split the shape into rectangles, triangles, and circles, add the areas, and subtract areas of holes or overlaps.

When every length is multiplied by k, perimeters scale by k and areas scale by k².

Use consistent units: 1 m = 100 cm, 1 m² = 10 000 cm², 1 hectare = 10 000 m².

Mean = sum of all values ÷ number of values; it is affected by extreme values, unlike the median.

Median: arrange the data in order; take the middle value, or the mean of the two middle values when the count is even.

Mode is the most frequent value; a data set may have several modes or none.

Range = largest value − smallest value; it measures spread but ignores all middle values.

In a bar chart or line chart, read values from the axis scale; the height of a bar is proportional to the frequency it represents.

Probability of an event = number of favourable outcomes ÷ number of equally likely outcomes, so 0 ≤ P ≤ 1.

P(not A) = 1 − P(A); for mutually exclusive events P(A or B) = P(A) + P(B).

Relative frequency = number of occurrences ÷ total number of trials; it estimates probability and approaches it as trials increase.

Modeling steps: name the unknown with a variable, translate each condition into an equation or inequality, solve, then check the answer against the original wording and units.

Percent problems: part = percent × whole; a price increased by p% becomes the original multiplied by (1 + p/100).

Motion problems use distance = speed × time; for two objects moving toward each other, their speeds add; in the same direction, they subtract.

Work problems: if one worker finishes a job in a hours, the rate is 1/a of the job per hour; combined rates add.

Mixture problems track the amount of a substance: (concentration₁ × volume₁) + (concentration₂ × volume₂) = final concentration × total volume.

To prove a statement false, one counterexample suffices; to prove it true, a general argument covering all cases is needed.

Justify each step by naming the property or theorem used (distributive law, angle sum of a triangle, congruence test) rather than only stating the result.

Answers to word problems must satisfy the context: lengths and counts cannot be negative, and quantities are usually whole numbers.