Definitions
Fraction
A mathematical way to represent a part of a whole number by using two numbers, where the top number (numerator) denotes how many parts are taken, and the bottom number (denominator) indicates into how many parts the whole is divided.
Straight Line Graph
A graph representing a linear equation in two dimensions, where the graph is a straight line, and each point on the line is a solution of the equation.
Equation
A mathematical statement that asserts the equality of two expressions, often using letters to represent unknowns.
Expression
A mathematical phrase that can contain numbers, variables, and operators, but does not have an equality sign.
Ratio
A relationship between two quantities, often represented as a:b indicating how many times the first number contains the second.
Bracket Expansion
The process of multiplying out the terms inside a bracket with the terms outside the bracket(s), applicable to single and double brackets.
Factorisation
The process of breaking down an expression into a product of simpler expressions or factors.
Inequality
A relation between two expressions that may not be equal, expressed using symbols such as <, >, ≤, or ≥.
Sequence
An ordered list of numbers following a specific pattern.
Nth Term Rule
A formula that allows the calculation of the nth term of a sequence using its position number n.
Fractions
Fractions are a key concept in mathematics, used to denote parts of a whole. For example, 3/4 represents three parts out of four. Operations involving fractions include addition, subtraction, multiplication, and division. Consider adding the fractions 1/2 and 1/3:
The common denominator of 2 and 3 is 6. Convert each fraction to have this common denominator. Thus, 1/2 becomes 3/6 and 1/3 becomes 2/6. Adding these, we have 3/6 + 2/6 = 5/6.
Straight Line Graphs
Straight line graphs visually represent linear equations. The standard form is y = mx + c, where m is the slope and c is the y-intercept. For example, in the equation y = 2x + 1, the graph is a line with slope 2 passing through the point (0,1) on the y-axis.
Solving Equations
Solving equations involves finding the values of variables that make the equation true. For example, solve the equation 2x + 3 = 11:
Subtract 3 from both sides: 2x = 8.
Divide both sides by 2: x = 4.
Simplifying Expressions
Simplifying expressions involves combining like terms and performing operations to make the expression simpler. For example, simplify 3x + 5x - 2:
Combine like terms: (3x + 5x) - 2 = 8x - 2.
Ratio
Ratios compare two quantities. For example, the ratio of 8 to 4 is written as 8:4, which simplifies to 2:1. This indicates that for every 2 units of the first quantity, there is 1 unit of the second quantity.
Expanding Single and Double Brackets
Expanding brackets means multiplying out expressions. For a single bracket, such as 3(x + 4), you perform: 3 * x + 3 * 4, resulting in 3x + 12.
For double brackets, such as (x + 2)(x + 3), use distributive law:
x(x + 3) + 2(x + 3) = x² + 3x + 2x + 6.
Combine like terms: x² + 5x + 6.
Factorising
Factorising is the reverse process of expanding. It involves writing an expression as a product of its factors. For example, factorise x² + 5x + 6:
The factors of 6 that add up to 5 are 2 and 3, so it becomes (x + 2)(x + 3).
Inequalities
Inequalities express a range of possible values for a variable. For example, x + 1 > 3:
Subtract 1 from both sides: x > 2.
This inequality means x can be any number greater than 2.
Generating a Sequence
A sequence is a set of numbers following a pattern. An example is an arithmetic sequence, where each term after the first is obtained by adding a constant called the common difference. For instance, the sequence 2, 5, 8, 11 adds 3 each time.
Finding the Nth Term Rule
The nth term rule provides a formula for the nth term of a sequence. For the arithmetic sequence 3, 5, 7, 9 with a common difference of 2, the nth term rule is 3 + (n - 1) * 2. Simplifying gives 2n + 1.
To remember :
The topics covered provide fundamental mathematical concepts and skills used across various domains. Fractions allow the representation of non-integer numbers. Straight line graphs visually express linear equations. Solving equations involves finding variable values. Expressions are simplified by combining like terms. Ratios compare relative quantities. Expanding brackets involves distribution of terms, while factorising is its inverse operation. Inequalities provide ranges of solutions. Sequences follow specific patterns, where the nth term formula offers a way to determine any term in a sequence. Mastering these topics builds a strong foundation in mathematical proficiency.