Maths revision Paper 2

Simplifying Expressions by Collecting Like Terms
- Combines terms that have the same variables and powers.
- Example: To simplify $3x + 2x$ , combine to get $5x$ .
Substituting into Expressions
- Involves replacing a variable in an expression with a specific value.
- Example: Given $x = 3$ , substitute into $2x + 5$ to get $2(3) + 5 = 11$ .
Writing Algebraic Proofs
- Establishes the validity of algebraic statements using logical reasoning.
- Must show each step and maintain equality.
Solving Equations
- Variable on Both Sides: Move variables to one side to solve.
- Example: $2x + 3 = x + 5$
  1. Move $x$ : $2x - x = 5 - 3$
  2. Simplify to get $x = 2$ .
- Two or More Steps: Requires multiple operations to isolate the variable.
- Example: For $3x + 5 = 20$ , subtract 5 and divide by 3.
Expanding Double Brackets
- Use the FOIL method (First, Outside, Inside, Last) to expand.
- Example: $(x + 2)(x + 3) = x^2 + 5x + 6$ .
Factorising Quadratic Expressions
- Writing quadratics as the product of their factors.
- Example: Factor $x^2 + 5x + 6$ to $(x + 2)(x + 3)$ .

Drawing and Interpreting Scale Diagrams
- Represents objects proportionally.
- Important for visualizing real-world dimensions.
Calculating Bearings
- Defined as the direction of one point from another.
- Typically measured clockwise from the North direction in degrees.
Finding the Area of Triangles
- Area = $\frac{1}{2} \times base \times height$ .
- Example: For a triangle with base 5 and height 3, Area = $\frac{1}{2} \times 5 \times 3 = 7.5$ .
Line and Shape Properties
- Understanding the characteristics of different geometric shapes.
- Examples include properties of triangles, quadrilaterals, and circles.
Finding the Circumference of Circles
- Circumference = $2 \pi r$ , where $r$ is the radius.
- Example: For a circle of radius 4, Circumference = $2 \pi \times 4 = 8\pi$ .

Rotation
- Involves turning a shape around a fixed point.
- Defined by angle and direction (clockwise or anti-clockwise).
Translation
- Moving a shape without rotating or flipping.
- Defined by a vector that indicates the distance and direction.

Using Pythagoras' Theorem in 2D
- Formula: $a^2 + b^2 = c^2$ , where $c$ is the hypotenuse.
- Used to find unknown lengths in right-angled triangles.
Finding Unknown Angles in Right Angled Triangles
- Involves using trigonometric ratios: Sine, Cosine, and Tangent.
- Example: $\sin(\theta) = \frac{opposite}{hypotenuse}$ .
Finding Unknown Sides in Right Angled Triangles
- Apply Pythagoras' theorem or trigonometric ratios based on known values.
Using a Calculator
- Important for calculations involving functions and equations.
- Must be familiar with how to input expressions correctly.

Converting Between Fractions, Decimals, and Percentages
- Critical for solving many math problems.
- Example: Convert $\frac{1}{2}$ to decimal (0.5) and percentage (50%).
Finding the Lowest Common Multiple (LCM)
- The smallest common multiple of two or more numbers.
- Example: LCM of 4 and 5 is 20.
Finding Prime Numbers
- Numbers greater than 1 with no divisors other than 1 and themselves.
- Example: 2, 3, 5, 7, etc. are prime numbers.
Index Rules with Positive Indices
- Laws governing the operations of exponents.
- Examples: $a^m \times a^n = a^{m+n}$ , $\frac{a^m}{a^n} = a^{m-n}$ .
Calculating with Roots and Powers
- Operations involving square roots and powers.
- Example: $\sqrt{16} = 4$ , $2^3 = 8$ .
Rounding Decimals
- Techniques to round to a specific number of decimal places.
- Example: Round 3.456 to 3.46 (to 2 decimal places).
Solving Direct Proportion Word Problems
- Problems where two quantities increase or decrease together.
- Example: If 3 apples cost $1, then 6 apples cost $2.
Calculating with Rates
- Determining unit rates or rates of change in problems.
Percentage Change with a Calculator
- Formula: $\text{Percentage Change} = \frac{new - old}{old} \times 100$ .
- Example: If a price increases from $50 to $60, the change is $\frac{60 - 50}{50} \times 100 = 20\%$ .
Finding the Percentage an Amount Has Been Changed By
- Requires percentage formula knowledge.

Convert Between Ratios, Fractions, and Percentages
- Essential for comparative problems.
- Example: A ratio of 2:3 can be expressed as $\frac{2}{5}$ or 40%.
Write Probabilities as Fractions
- Probability defined as the number of successful outcomes over total outcomes.
- Example: Probability of rolling a 3 on a die is $\frac{1}{6}$ .
Drawing Pie Charts
- Visual representation of data in proportions.
- Each slice represents a percentage of the whole.
Drawing Line Graphs
- Displays data points over time or a specific interval.
Interpreting Line Graphs
- Ability to extract information from graph trends.
Frequency Trees
- Diagrams to display probabilities and combinations.
Calculating the Mean
- The average of a data set, calculated by adding all values and dividing by the count.
- Example: For data set {3, 4, 5}, Mean = $\frac{3 + 4 + 5}{3} = 4$ .