Japan 中学2年 Mathematics

Chapters: 4

1. Numbers and expressions

Expressions with letters · Simultaneous linear equations

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.
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.

Properties of basic plane figures · Congruence and proof

Lines and Angles: Linear Pair, Vertically Opposite and Parallel Lines – An angle is the turn between two rays that start from the same point. Angles on a straight line add up to 180° (linear pair). When two lines cross, the opposite angles are equal. When a transversal cuts two parallel lines, corresponding angles are equal, alternate interior angles are equal, and interior angles on the same side add up to 180°. The reverse is also true, and we can prove such facts by contradiction: assume the opposite and show it leads to something impossible.
Congruence of Triangles – Two triangles are congruent when one can be placed exactly on top of the other: all three sides and all three angles match. You do not need to check all six parts. Any one of SAS, SSS, ASA, AAS or RHS is enough. SSA is not enough. In an isosceles triangle the angles opposite the equal sides are equal, and the converse is also true.

Linear functions

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.

Data distribution · Probability

Introduction to Probability: Scale, Experiments, Sample Spaces and Trees – Probability is a number from 0 to 1 that tells how likely something is. 0 means it can never happen, 1 means it will surely happen. We can find it by doing an experiment many times (empirical probability), or by listing every possible result (the sample space) and counting the ones we want. Tree diagrams and tables help us list results when two things happen together.