Poland Szkoła podstawowa, klasa VII Mathematics

Chapters: 7

1. Powers with rational bases

Powers with natural exponents · Scientific notation

Laws of Exponents – An exponent tells how many times a base is multiplied by itself. Same base: multiply → add exponents, divide → subtract, power of a power → multiply. a⁰ = 1, a⁻ⁿ = 1/aⁿ and a^(1/n) is the nth root.
Scientific Notation (Standard Form) – Scientific notation writes any number as a × 10ⁿ, where 1 ≤ a < 10 and n is a whole number (an integer). For big numbers the decimal point moves left and n is positive. For small numbers it moves right and n is negative. To multiply, multiply the a's and add the powers; to divide, divide the a's and subtract the powers; then fix a so it is between 1 and 10. Unit prefixes like kilo (10³), mega (10⁶), milli (10⁻³), micro (10⁻⁶) and nano (10⁻⁹) are powers of ten with names.

Square and cube roots · Operations on roots

Square Roots: Build a Square, Find Its Side – The square root of a number n is the number that, multiplied by itself, gives n. √25 = 5 because 5 × 5 = 25. Picture n tiles arranged in a square: the root is the side. Perfect squares (1, 4, 9, 16, …) have whole-number roots. Other roots are irrational and lie between two whole numbers; we estimate them or simplify them, like √50 = 5√2.

Expressions with one or more variables

Algebraic Expressions: Variables, Terms and Simplifying – An algebraic expression uses letters (variables) and numbers joined by +, −, × and ÷, like 3x + 2. A variable stands for a number that can change. An expression is made of terms (3x and 2). In 3x, 3 is the coefficient; a term with no letter, like 2, is the constant. Like terms have exactly the same letter part (3x and 5x) and can be added; unlike terms (3x and 2, or x and x²) cannot. Substitution means putting a number in place of the letter to find the value. Expanding removes brackets: a(b + c) = ab + ac. Factorising is the reverse: take out the common factor. A formula is an expression that gives one quantity from others, and an identity is true for every value of the letter.

Monomials and algebraic sums

Multiplying Polynomials – To multiply polynomials, multiply every term of the first by every term of the second (the distributive law), then collect like terms. Coefficients multiply; powers of the same letter add (x² · x³ = x⁵). A rectangle of area tiles shows every product: (x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6. To divide a polynomial by a monomial, divide each term by it.

Percentages

Percentages – Per cent means "out of 100", so x% = x/100. To find x% of an amount, multiply by x/100. Percentage change = (change ÷ original) × 100. To increase by r%, multiply by (1 + r/100); to decrease by r%, multiply by (1 − r/100). Successive changes multiply their multipliers. For reverse percentage, divide the new value by the multiplier.

Solving linear equations · Applying equations

Linear Equations in One Variable – A linear equation in one variable has one unknown (like x) with power 1, for example 2x + 3 = 11. An equation is like a balance: both sides are equal. To solve it, do the same thing to both sides (add, subtract, multiply or divide by the same non-zero number) until x is alone. Always check by putting the answer back in.

Directly proportional quantities

Ratio and Proportion – A ratio a : b compares two amounts of the same kind. Divide both parts by the same number to simplify it. To share an amount in a ratio, add the parts, find one part, then multiply. Two quantities are in direct proportion when y = k·x (double one, double the other) and in inverse proportion when x·y = k (double one, halve the other).