France Troisième Mathematics
Chapters: 5
1. Numbers and calculations
Irreducible fractions, multiples and divisors · Powers with any base and negative exponents · Using square roots in problems · Double distributivity · Difference of squares, zero-product equations, x² = a
- 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.
- 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.
- 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.
- 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.
- Quadratic Equations – A quadratic equation has x² as its highest power: ax² + bx + c = 0 with a ≠ 0. It has at most two roots. Find them by factorising (split the middle term, then set each bracket to zero) or by the formula x = (−b ± √(b² − 4ac)) / 2a. The discriminant D = b² − 4ac tells the nature of roots: D > 0 two different real roots, D = 0 two equal roots, D < 0 no real roots.
2. Data, probability and functions
Statistics: range and histograms · Probability: frequencies stabilise, simulation · Percentage change and multiplier · Functions: variable, image, preimage, notation · Linear and affine functions
- 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.
- 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.
- 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.
3. Quantities and measures
Volume of a sphere · Effect of transformations on lengths, areas, volumes
- Surface Areas and Volumes: Combined Solids – Many real objects are two simple solids stuck together, like a cone on a cylinder. The surface area is only the outside skin you can touch, so the hidden joint is left out. The volume is the space inside, so you simply add the volumes of the parts.
- 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.
4. Space and geometry
Similar triangles · Intercept theorem (butterfly case) and converse · Right-triangle trigonometry: cos, sin, tan · Rotations and dilations · Latitude and longitude; sections of solids
- 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 3: nested loops, custom blocks
- 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.