Russia 7 класс Algebra (advanced)
Chapters: 4
1. Numbers and calculations
Rational numbers · Divisibility
- Rational Numbers: Number Line, Density and Decimals – A rational number is any number that can be written as p/q, where p and q are integers and q is not 0. Examples: 3/4, −5/3, 7 (= 7/1), 0.25 (= 1/4). To place p/q on the number line, cut each unit into q equal parts and hop p parts from 0. Between any two rational numbers there is always another one (their average), so there are endlessly many between them. The decimal of a rational number either ends (terminating) or repeats a block for ever (non-terminating repeating).
- Divisibility: Multiples, Divisors, GCD and LCM – a is divisible by b when a = b × k for a whole number k: b is a divisor (factor) of a and a is a multiple of b. Every whole number a can be written as a = b × q + r with 0 ≤ r < b (division with remainder); b divides a exactly when r = 0. Quick tests tell divisibility by 2, 3, 4, 5, 6, 8, 9, 10 and 11 from the digits. The GCD is the greatest common divisor, found by prime factors or by Euclid's algorithm; LCM is the least common multiple, and GCD × LCM = a × b. Two numbers are coprime when their GCD is 1.
2. Algebraic expressions
Identities · Polynomials · Special products · Factorisation
- 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.
- Introduction to Polynomials – An algebraic expression is made of terms like 3x², −5x and 7. It is a polynomial when every power of the variable is a whole number (0, 1, 2, …). The degree is the biggest power. Degree 1 polynomials, y = ax + b, are called linear. They model things that grow or shrink by the same amount each step. a is the slope (change per step) and b is the y-intercept (starting value).
- Exploring Algebraic Identities – An identity is an equation that is true for every value of the letters. Pictures prove them: a square of side (a + b) splits into a², ab, ab and b², so (a + b)² = a² + 2ab + b². In the same way we get (a − b)², a² − b² = (a + b)(a − b), (x + a)(x + b), (a + b + c)² and (a + b)³. Read backwards, identities help us factorise, calculate fast and simplify rational expressions.
3. Equations and inequalities
Linear equations · Systems of linear 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.
- Pair of Linear Equations in Two Variables – Two equations like a₁x + b₁y = c₁ and a₂x + b₂y = c₂ each make a straight line. The answer that fits both is the point where the lines meet. Lines that cross give one answer, parallel lines give none, and lines that lie on top of each other give endless answers. We can find the answer by drawing (graph), by substitution or by elimination.
4. Functions
Function concept · Linear function
- Functions: Input, Rule, Output – A function is a rule that gives exactly one output for each input. We write it as f(x). The output f(a) is called the image of a; an input that gives a certain output is a preimage. A function can be shown as words, a table, a formula or a graph. Its graph can go up (increasing), go down (decreasing) and have a highest point (maximum) or lowest point (minimum).
- Linear Functions – A linear function adds the same amount to y every time x goes up by 1. Its rule is y = mx + c (also written f(x) = mx + b). m is the slope: rise ÷ run, the change in y for each 1 step in x. c is the y-intercept: the value of y when x = 0. Its graph is always a straight line.