National Year 11 Mathematics
Chapters: 4
1. 3.3 Ratio, proportion and rates of change
Units, scales and ratio · Proportion and percentages · Rates of change
- 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).
- Compound Interest – Interest is money paid for using money. Simple interest is paid only on the starting amount (the principal), so it grows by the same step every year. Compound interest is paid on the principal plus all the interest earned so far, so the steps get bigger: interest earns interest. The amount after n years is A = P(1 + r/100)ⁿ. If interest is added k times a year, use rate r/k and k × n periods: A = P(1 + r/(100k))^(kn). Adding it every instant gives continuous compounding, A = Pe^(rt). The same idea run backwards gives depreciation: A = P(1 − r/100)ⁿ. Compound growth is exponential growth.
2. 3.4 Geometry and measures
3.4.1 Properties and constructions · 3.4.2 Mensuration and calculation · 3.4.3 Vectors
- Circle Theorems – Circle theorems are a small set of angle rules that are always true inside a circle. The angle at the centre is twice the angle at the edge. The angle in a semicircle is 90°. Angles in the same segment are equal. Opposite angles of a cyclic quadrilateral add to 180°. A tangent meets the radius at 90°, two tangents from one point are equal, and the tangent–chord angle equals the angle in the alternate segment.
- 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.
- Vectors: Column Vectors, Adding, Scaling and Proofs – A vector has a size and a direction. On a grid we write it as a column vector (across over up). Add vectors tip to tail by adding their parts; multiply by a number to stretch or reverse them. Its length is found with Pythagoras. In geometry, we write any path as a sum of known vectors to prove lines are parallel or points are midpoints.
3. 3.5 Probability
Experimental and theoretical probability · Combined events
- 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.
- Combined Events, Tree Diagrams and Conditional Probability – When two things happen, list every outcome in a sample space, a table, a Venn diagram or a tree diagram. Multiply along tree branches (the multiplication rule) and add the paths you want. If the first event changes the second, the events are dependent: picking without replacement is the classic case. Conditional probability P(A | B) is the chance of A when we already know B happened: P(A | B) = P(A and B) ÷ P(B).
4. 3.6 Statistics
Sampling and representing data · Analysing data
- 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.