France Quatrième Mathematics
Chapters: 5
1. Numbers and calculations
Signed decimals: multiplication and division · Rational numbers: reciprocal, multiplying and dividing fractions · Square roots: introduction and perfect squares · Powers of ten, scientific notation, unit prefixes · Primes up to 100 and prime factorisation · Algebra: single distributivity, expand and factorise · Equations: unknown, solution, solving first-degree equations
- Integers and Signed Numbers – Integers are the whole numbers and their negatives: …, −3, −2, −1, 0, 1, 2, 3, …. On a number line, numbers grow to the right. Adding a positive moves right, adding a negative moves left. Subtracting a number is the same as adding its opposite. For × and ÷, same signs give a positive answer and different signs give a negative answer. The same sign rules work for signed decimals like −2.5.
- 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).
- 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.
- 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.
- Fundamental Theorem of Arithmetic – Every whole number bigger than 1 is either a prime or can be written as a product of primes in exactly one way (only the order can change). This is the Fundamental Theorem of Arithmetic. Using these prime "building blocks": HCF = product of the smallest powers of the common primes; LCM = product of the greatest powers of all primes. For two numbers, HCF × LCM = a × b.
- 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.
- 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.
2. Data, probability and functions
Statistics: median, counts and frequencies · Probability: events and complementary events · Proportionality: fourth proportional and graphs · Relationship between two quantities: table, formula, graph
- Statistics: Mean, Median and Mode of Grouped Data – When data is put into classes (like marks 20–30), we cannot see each value. We use the middle of each class to find the mean, the running total to find the median, and the tallest bar to find the mode. All three tell us the 'centre' of the data in different ways.
- Probability: Chances of a Single Event – Probability tells how likely something is, as a number from 0 to 1. When all outcomes are equally likely, P(event) = number of favourable outcomes ÷ total number of outcomes. An impossible event has probability 0 and a sure event has probability 1.
- 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).
- 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).
3. Quantities and measures
Volume of pyramid and cone · Compound quantities and units · Effect of enlargement or reduction on length, area, volume
- Surface Area and Volume of Cuboid, Cylinder, Cone, Pyramid and Sphere – Surface area is the total area of the outside skin of a solid, like the paper needed to wrap it. Volume is the space inside, like the water it can hold. A cuboid's volume is the number of 1 cm cubes that fit in it. A cone holds one third of a cylinder with the same base and height, and a pyramid holds one third of the matching prism.
- Compound Units: Speed, Density, Unit Price and Other Rates – A compound unit joins two units, like metres per second (m/s) or grams per cubic centimetre (g/cm³). "Per" means "in each one", so a rate tells you how much of one amount fits into ONE unit of another. Speed = distance ÷ time, density = mass ÷ volume, unit price = cost ÷ amount. To convert a compound unit, convert the top unit and the bottom unit separately.
- Similar Triangles – Two figures are similar when they have the same shape but maybe a different size. Two triangles are similar if their matching angles are equal and their matching sides are in the same ratio. Basic Proportionality Theorem (BPT): a line parallel to one side of a triangle cuts the other two sides in the same ratio; its converse is also true. You can prove similarity with AA, SSS or SAS.
4. Space and geometry
Pythagoras theorem, converse and contrapositive · Intercept theorem (nested triangles) and converse · Cosine in a right triangle · Translations: friezes and tilings · Coordinates in a cuboid; nets of pyramids and cones
- Pythagoras Theorem – In any right-angled triangle, the square on the longest side (the hypotenuse) equals the sum of the squares on the other two sides: a² + b² = c².
- Similar Triangles – Two figures are similar when they have the same shape but maybe a different size. Two triangles are similar if their matching angles are equal and their matching sides are in the same ratio. Basic Proportionality Theorem (BPT): a line parallel to one side of a triangle cuts the other two sides in the same ratio; its converse is also true. You can prove similarity with AA, SSS or SAS.
- Trigonometric Ratios (sin, cos, tan) – In a right triangle, pick one sharp (acute) angle θ. The side facing θ is the opposite, the side touching θ (not the longest) is the adjacent, and the longest side is the hypotenuse. sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. cosec, sec and cot are their flips. The ratios depend only on the angle, not on the size of the triangle. Learn the table for 0°, 30°, 45°, 60°, 90°.
- Transformations: Slide, Flip, Turn and Resize a Shape – A transformation moves a shape to a new place. Translation slides it, reflection flips it in a mirror line, rotation turns it about a centre, enlargement changes its size from a centre. The first three keep size and shape (congruent image). Enlargement keeps shape but changes size (similar image). Each has a simple coordinate rule.
- Solid Geometry: Points, Lines and Planes in Space – Solid geometry studies figures in three dimensions. Three points not on one line fix a plane. Two lines in space can be parallel, intersecting or skew (not in one plane). A line can lie in a plane, cut it, or be parallel to it; it is perpendicular to a plane if it is perpendicular to two intersecting lines of that plane. Angles in space are found by projecting onto a plane and using right triangles.
5. Algorithms and programming
Block programming level 2: variables, conditions, events
- Block Programming: Algorithms, Loops, Variables, Events and Sensors – In block programming you snap coloured blocks together to build a program. An algorithm is the plan; the program is the plan written in blocks. Loops repeat blocks, variables store changing values, conditions choose between actions, events start the program, and sensors feed the robot facts about the world. Custom blocks (subprograms) give a name to a group of blocks so you can reuse it.