Mathematics (KS3)
UK Key Stage 3 mathematics: algebra, geometry, number, ratio and proportion, statistics and probability.
Ämne: Matematik · Nivå: Högstadium (13–15) · 410 kort
Innehåll
- The order of operations is BIDMAS: Brackets, Indices, Division and Multiplication (left to right), Addition and Subtraction (left to right).
- A prime number has exactly two distinct factors: 1 and itself. The smallest prime is 2, which is also the only even prime.
- The Highest Common Factor (HCF) of two numbers is the largest number that divides both exactly. The Lowest Common Multiple (LCM) is the smallest positive number that both divide into exactly.
- When multiplying two negative numbers, the result is positive. When multiplying a negative by a positive, the result is negative.
- Standard form (scientific notation) writes a number as A x 10^n, where 1 <= A < 10 and n is an integer. Example: 47000 = 4.7 x 10^4.
- To add or subtract fractions, you must first rewrite them with a common denominator. Example: 1/3 + 1/4 = 4/12 + 3/12 = 7/12.
- To divide one fraction by another, multiply by its reciprocal: a/b ÷ c/d = a/b × d/c.
- A percentage is a fraction out of 100. 25% = 25/100 = 1/4 = 0.25.
- Like terms have exactly the same variable parts (including powers). 3x and 5x are like terms; 3x and 3x^2 are not.
- To expand a single bracket, multiply each term inside the bracket by the term outside: 3(x + 4) = 3x + 12.
- FOIL stands for First, Outer, Inner, Last - a method for expanding two binomial brackets. (x + 2)(x + 3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6.
- The nth term of a linear sequence has the form an + b, where a is the common difference and b is chosen so the first term is correct.
- Angles on a straight line add up to 180°. Angles around a point add up to 360°. Angles in a triangle add up to 180°. Angles in a quadrilateral add up to 360°.
- When two parallel lines are cut by a transversal: corresponding angles are equal, alternate angles are equal, and co-interior angles add up to 180°.
- Pythagoras' theorem: in a right-angled triangle, a^2 + b^2 = c^2, where c is the hypotenuse (the side opposite the right angle).
- Area of a triangle = (1/2) × base × perpendicular height. Area of a parallelogram = base × perpendicular height. Area of a trapezium = (1/2)(a + b)h.
- The area of a circle is πr^2 and the circumference is 2πr (or πd, where d is the diameter).
- The sum of interior angles of an n-sided polygon is (n - 2) × 180°. Each exterior angle of a regular polygon is 360°/n.
- Volume of a cuboid = length × width × height. Volume of a prism = cross-sectional area × length. Volume of a cylinder = πr^2 × h.
- Congruent shapes are identical in size and shape. Similar shapes have the same angles and proportional sides but may be different sizes.
- Mean = sum of values ÷ number of values. Median = middle value when ordered. Mode = most common value. Range = highest - lowest.
- A scatter graph can show correlation: positive (both increase), negative (one increases as the other decreases), or none (no clear pattern).
- Probabilities are between 0 (impossible) and 1 (certain), and can be expressed as fractions, decimals or percentages. The probabilities of all possible outcomes sum to 1.
- The equation of a straight line is y = mx + c, where m is the gradient (slope) and c is the y-intercept (where the line crosses the y-axis).
- A ratio a:b can be simplified by dividing both parts by their HCF. To share a quantity in a given ratio, divide by the total number of parts and multiply by each share.
- An inequality uses <, >, ≤, or ≥ instead of =. Solve it like an equation, but if you multiply or divide by a negative number, flip the inequality sign.
- A prime factor tree breaks a number down into primes by repeatedly splitting it into factors until only primes remain. Example: 60 = 2 x 2 x 3 x 5 = 2^2 x 3 x 5.
- A terminating decimal has a finite number of digits after the decimal point (e.g. 0.25). A recurring decimal repeats a pattern forever (e.g. 0.333... = 1/3). Any fraction with a denominator made only of 2s and 5s terminates.
- Index laws: a^m x a^n = a^(m+n); a^m / a^n = a^(m-n); (a^m)^n = a^(mn); a^0 = 1 for any non-zero a; a^(-n) = 1/a^n.
- The reciprocal of a number n is 1/n. The reciprocal of a fraction a/b is b/a. A number multiplied by its reciprocal gives 1. Zero has no reciprocal.
- To rationalise a denominator of the form 1/sqrt(a), multiply numerator and denominator by sqrt(a). For example, 1/sqrt(2) = sqrt(2)/2.
- A recurring decimal can be converted into a fraction using algebra. If x = 0.444..., then 10x = 4.444..., so 9x = 4 and x = 4/9.
- To change the subject of a formula, use inverse operations on both sides until the desired letter is alone. Example: from v = u + at, make a the subject: a = (v - u)/t.
- To factorise a quadratic of the form x^2 + bx + c, find two numbers that multiply to give c and add to give b. Example: x^2 + 5x + 6 = (x + 2)(x + 3).
- A geometric sequence multiplies by a constant ratio between terms (e.g. 2, 6, 18, 54 with ratio 3). A Fibonacci-like sequence adds the previous two terms (e.g. 1, 1, 2, 3, 5, 8).
- A quadratic sequence has a constant second difference. The first differences increase or decrease by the same amount each step. Example: 3, 6, 11, 18 has first differences 3, 5, 7 and second difference 2.
- A bearing is an angle measured clockwise from north, given as a three-figure number between 000 and 360 degrees. Example: due east is 090 and due south is 180.
- To construct a perpendicular bisector of a line segment, set compasses to more than half the length, draw arcs from each end above and below, then join the two crossing points. Every point on the bisector is equidistant from the segment's endpoints.
- To construct an angle bisector, place the compass point on the vertex and draw an arc cutting both rays. From each intersection, draw arcs of equal radius that cross. Join the vertex to the crossing point - this line bisects the angle.
- A locus is the set of all points satisfying a rule. The locus of points a fixed distance from a single point is a circle; the locus equidistant from two points is the perpendicular bisector of the segment joining them.
- The four main transformations are: reflection (over a mirror line), rotation (about a centre by an angle), translation (shift by a column vector) and enlargement (scaled by a factor from a centre).
- A column vector (a; b) means move a units right and b units up. Negative components mean left or down. Adding two column vectors adds their components separately.
- Arc length = (theta/360) x 2 pi r and sector area = (theta/360) x pi r^2, where theta is the sector's central angle in degrees and r is the radius.
- Circle theorems include: the angle at the centre is twice the angle at the circumference (when standing on the same arc), angles in the same segment are equal, and opposite angles of a cyclic quadrilateral add up to 180 degrees.
- Volume of a pyramid = (1/3) x base area x perpendicular height. Volume of a cone = (1/3) x pi r^2 x h. Volume of a sphere = (4/3) x pi r^3.
- The surface area of a cylinder (closed) is 2 pi r^2 + 2 pi r h: two circular ends plus a rectangle (the unrolled curved surface) with width equal to the circumference.
- A net is a 2D pattern that folds into a 3D shape. A cube has 11 distinct nets, all made of six connected squares.
- Trigonometry in a right-angled triangle uses SOH CAH TOA: sin(theta) = opposite/hypotenuse, cos(theta) = adjacent/hypotenuse, tan(theta) = opposite/adjacent.
- In a stem-and-leaf diagram, the stem holds the larger place value (e.g. tens) and the leaves hold the next digit (e.g. units). The leaves must be ordered and a key is required.
- To estimate the mean from a grouped frequency table, multiply each midpoint by its frequency, sum these products, then divide by the total frequency.