## Number Basics

Numbers are fundamental in maths. Integers are whole numbers (positive, negative, or zero). Prime numbers are integers greater than 1 with exactly two factors: 1 and themselves (e.g., 2, 3, 5, 7, 11). Square numbers are the result of multiplying an integer by itself (e.g., 1, 4, 9, 16). Cube numbers are the result of multiplying an integer by itself three times (e.g., 1, 8, 27).

A factor divides a number exactly (e.g., factors of 12 are 1, 2, 3, 4, 6, 12). A multiple is a number in another number's times table (e.g., multiples of 5 are 5, 10, 15...). The Highest Common Factor (HCF) is the largest factor shared by two or more numbers. The Lowest Common Multiple (LCM) is the smallest multiple shared by two or more numbers.

## Operations and Order

Always follow the order of operations, often remembered as BIDMAS or BODMAS:

• Brackets
• Indices (Powers/Roots) / Orders
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

Operations with negative numbers:

• Adding a negative is subtracting: 5 + (-3) = 2
• Subtracting a negative is adding: 5 - (-3) = 8
• Multiplying/dividing: same signs give positive, different signs give negative.

## Fractions, Decimals, and Percentages

These are different ways to represent parts of a whole.

• Fractions: Can be simplified by dividing the numerator and denominator by their HCF. To add/subtract, find a common denominator. To multiply, multiply numerators and denominators. To divide, "flip" the second fraction and multiply.
• Decimals: Place value is key (tenths, hundredths, etc.). To convert a fraction to a decimal, divide the numerator by the denominator.
• Percentages: "Per cent" means "out of 100". To convert a percentage to a decimal, divide by 100 (e.g., 25% = 0.25). To find a percentage of an amount, convert the percentage to a decimal or fraction and multiply.

## Rounding and Estimation

Rounding makes numbers simpler.

• To a decimal place (d.p.): Look at the digit after the required d.p. If it's 5 or more, round up; otherwise, keep it the same.
• To significant figures (s.f.): The first significant figure is the first non-zero digit. Apply the same rounding rule.

Estimation involves rounding numbers in a calculation to 1 significant figure before performing the operation. This gives a rough answer to check if your calculator answer is reasonable.

## Ratio

Ratio compares quantities. Ratios can be simplified by dividing all parts by their HCF (e.g., 10:15 simplifies to 2:3). To share an amount in a given ratio, add the parts of the ratio, divide the total amount by this sum to find the value of one part, then multiply by each part of the ratio.

## Understanding Fractions

A fraction represents a part of a whole. It has a numerator (top number) and a denominator (bottom number).

• To simplify a fraction, divide both the numerator and denominator by their highest common factor. E.g., 4/8 simplifies to 1/2.
• Equivalent fractions represent the same value, e.g., 1/2 = 2/4 = 5/10.
• To add or subtract fractions, they must have a common denominator. Find the lowest common multiple (LCM) of the denominators, then adjust the numerators. E.g., 1/2 + 1/3 = 3/6 + 2/6 = 5/6.
• To multiply fractions, multiply the numerators and multiply the denominators. E.g., 1/2 * 3/4 = 3/8.
• To divide fractions, "Keep, Change, Flip" (KCF). Keep the first fraction, change division to multiplication, and flip the second fraction (find its reciprocal). E.g., 1/2 ÷ 1/4 = 1/2 * 4/1 = 4/2 = 2.
• To find a fraction of an amount, divide the amount by the denominator and multiply by the numerator. E.g., 2/3 of 30 is (30 ÷ 3) * 2 = 10 * 2 = 20.

## Decimals

Decimals are another way to represent parts of a whole, based on powers of 10.

• Place value is crucial: 0.1 (tenths), 0.01 (hundredths), 0.001 (thousandths).
• To compare decimals, look at the digits from left to right, starting with the largest place value.
• To add or subtract decimals, line up the decimal points.
• To multiply decimals, ignore the decimal points, multiply the numbers, then count the total number of decimal places in the original numbers and put the decimal point back in your answer. E.g., 0.2 * 0.3 = 0.06 (one decimal place + one decimal place = two decimal places).
• To divide decimals, make the divisor a whole number by multiplying both numbers by a power of 10. E.g., 6 ÷ 0.2 = 60 ÷ 2 = 30.
• Rounding decimals involves looking at the digit to the right of the desired place value. If it's 5 or more, round up; if less than 5, keep the digit the same.

## Percentages

A percentage means "out of 100". The symbol is %.

• To convert a percentage to a decimal, divide by 100. E.g., 25% = 0.25.
• To convert a decimal to a percentage, multiply by 100. E.g., 0.75 = 75%.
• To convert a percentage to a fraction, write it over 100 and simplify. E.g., 50% = 50/100 = 1/2.
• To convert a fraction to a percentage, convert to a decimal first (divide numerator by denominator), then multiply by 100. E.g., 1/4 = 0.25 = 25%.
• To find a percentage of an amount:
• Non-calculator: Find 10%, 5%, 1% and combine. E.g., 30% of 60: 10% of 60 = 6, so 30% = 3 * 6 = 18.
• Calculator: Convert the percentage to a decimal and multiply. E.g., 30% of 60 = 0.30 * 60 = 18.
• Percentage increase/decrease:
• Find the percentage amount, then add/subtract it from the original amount.
• Or use a multiplier: For a 10% increase, multiply by 1.10. For a 10% decrease, multiply by 0.90.
• To express one number as a percentage of another: (part / whole) * 100. E.g., 15 out of 20 = (15/20) * 100 = 75%.

## Ratio

Ratio is used to compare quantities of the same type. For example, if there are 3 red balls and 2 blue balls, the ratio of red to blue is 3:2.

To simplify a ratio, divide all parts by their highest common factor (HCF).

• Example: Simplify 10:15. The HCF of 10 and 15 is 5. Divide both by 5 to get 2:3.

To divide a quantity in a given ratio, first find the total number of parts in the ratio. Then divide the quantity by this total to find the value of one part. Finally, multiply the value of one part by each number in the ratio.

• Example: Divide £60 in the ratio 2:3. Total parts = 2 + 3 = 5. Value of one part = £60 / 5 = £12. So, 2 parts = 2 x £12 = £24, and 3 parts = 3 x £12 = £36.

## Proportion

Direct Proportion means that as one quantity increases, the other quantity increases at the same rate. If two quantities are directly proportional, their ratio is constant.

• Example: If 3 apples cost £1.50, how much do 5 apples cost? Find the cost of one apple: £1.50 / 3 = £0.50. Then multiply by 5: £0.50 x 5 = £2.50. This is often called the unitary method.

Best Buys problems involve comparing unit costs to find the cheapest option. Calculate the cost per unit (e.g., per gram, per item) for each option.

## Percentages

A percentage is a fraction out of 100.

To increase a quantity by a percentage, calculate the percentage amount and add it to the original quantity. Alternatively, multiply the original quantity by (1 + decimal equivalent of percentage).

• Example: Increase £80 by 10%. 10% of £80 = £8. £80 + £8 = £88. Or, £80 x 1.10 = £88.

To decrease a quantity by a percentage, calculate the percentage amount and subtract it from the original quantity. Alternatively, multiply the original quantity by (1 - decimal equivalent of percentage).

Simple Interest is calculated only on the original amount.

• Formula: Simple Interest = (Principal x Rate x Time) / 100

## Rates of Change

Rates of change compare how one quantity changes in relation to another. Common rates involve speed, density, and pressure.

• Speed, Distance, Time:
• Speed = Distance / Time
• Distance = Speed x Time
• Time = Distance / Speed
• Remember units must be consistent (e.g., km/h, km, h).
• Density, Mass, Volume:
• Density = Mass / Volume
• Mass = Density x Volume
• Volume = Mass / Density
• Pressure, Force, Area:
• Pressure = Force / Area
• Force = Pressure x Area
• Area = Force / Pressure

## Unit Conversions

Be able to convert between common units, especially for length (mm, cm, m, km), mass (g, kg), volume (ml, l), and time (seconds, minutes, hours, days).

• 1 km = 1000 m
• 1 m = 100 cm
• 1 cm = 10 mm
• 1 kg = 1000 g
• 1 litre = 1000 ml
• 1 hour = 60 minutes
• 1 minute = 60 seconds

## Algebra Basics and Equations

Algebra uses letters (called variables) to represent unknown numbers. It's like a puzzle where you figure out what the letters stand for. Understanding algebra is crucial for solving many mathematical problems.

## Simplifying Expressions

An expression is a group of terms without an equals sign (e.g., `3x + 5`).

• Collecting Like Terms: You can only add or subtract terms that are "alike". Like terms have the same letter raised to the same power.
• Example: `3a + 2b + a - b` simplifies to `4a + b`. (Here, `3a` and `a` are like terms; `2b` and `-b` are like terms).
• Multiplying Terms:
• Multiply the numbers first, then the letters.
• `3a × 2b = 6ab`
• `x × x = x²` (x squared)
• `4y × 2y = 8y²`
• Dividing Terms:
• Divide the numbers first, then the letters.
• `10x / 5 = 2x`
• `6ab / 2a = 3b`

## Substitution

Substitution means replacing letters in an expression with their given numerical values and then calculating the result.

• Example: If `a = 3` and `b = 5`, find the value of `2a + b`.
• `2a + b = (2 × 3) + 5 = 6 + 5 = 11`

## Solving Equations

An equation has an equals sign, and its goal is to find the value of the unknown variable. To solve an equation, you need to isolate the variable on one side of the equals sign. Remember to do the opposite operation to both sides to keep the equation balanced.

• One-step equations:
• `x + 5 = 12` (Subtract 5 from both sides) `x = 7`
• `3x = 15` (Divide by 3 on both sides) `x = 5`
• Two-step equations:
• `2x + 3 = 11`

1. Subtract 3 from both sides: `2x = 8`

2. Divide by 2 on both sides: `x = 4`

• Equations with unknowns on both sides (simple cases):
• `5x - 2 = 3x + 8`

1. Subtract `3x` from both sides: `2x - 2 = 8`

2. Add 2 to both sides: `2x = 10`

3. Divide by 2 on both sides: `x = 5`

## Forming Equations

You might need to translate a word problem into an algebraic equation.

• Example: "I think of a number, multiply it by 4, and add 7. The answer is 23."
• Let the number be `x`.
• Equation: `4x + 7 = 23`

## Angles

Angles are fundamental to geometry.

• An acute angle is less than 90°.
• An obtuse angle is between 90° and 180°.
• A reflex angle is greater than 180° but less than 360°.
• A right angle is exactly 90°.
• A straight angle is exactly 180°.

Key rules:

• Angles on a straight line add up to 180°.
• Angles around a point add up to 360°.
• Vertically opposite angles are equal.
• The sum of angles in a triangle is 180°.
• The sum of angles in a quadrilateral is 360°.

When parallel lines are intersected by a transversal:

• Corresponding angles are equal (F-shape).
• Alternate angles are equal (Z-shape).
• Interior (or co-interior) angles add up to 180° (C-shape).

## Perimeter and Area

Perimeter is the total distance around the outside of a 2D shape.

Area is the amount of surface a 2D shape covers.

Formulas to remember:

• Rectangle: Perimeter = 2(L + W), Area = Length × Width
• Triangle: Perimeter = sum of sides, Area = 0.5 × Base × Height
• Parallelogram: Area = Base × Height
• Trapezium: Area = 0.5 × (a + b) × Height (where a and b are the parallel sides)
• Circle:
• Circumference (perimeter) = πd (pi × diameter) or 2πr (2 × pi × radius)
• Area = πr² (pi × radius × radius)

Remember to use consistent units.

## Volume

Volume is the amount of 3D space an object occupies.

• Cuboid: Volume = Length × Width × Height
• Cylinder: Volume = πr²h (pi × radius² × height)

Units for volume are cubed (e.g., cm³, m³).

## Pythagoras' Theorem

This theorem applies only to right-angled triangles. It states that the square of the hypotenuse (the longest side, opposite the right angle) is equal to the sum of the squares of the other two sides.

• a² + b² = c² (where c is the hypotenuse).

## Measures and Conversions

It's crucial to know common units and how to convert between them.

• Length: 1 km = 1000 m, 1 m = 100 cm, 1 cm = 10 mm.
• Mass: 1 kg = 1000 g, 1 g = 1000 mg.
• Capacity: 1 litre = 1000 ml.

Always check the units required for your answer.

## Probability Basics

Probability measures the likelihood of an event happening, expressed as a number between 0 (impossible) and 1 (certain). It can also be written as a fraction, decimal, or percentage.

• Theoretical Probability: For equally likely outcomes, P(event) = (Number of favourable outcomes) / (Total number of possible outcomes).
• Experimental Probability (Relative Frequency): P(event) = (Number of times the event occurred) / (Total number of trials). This is an estimate and generally improves with more trials.
• Complementary Events: Two events are complementary if one or the other must happen, but not both. P(not A) = 1 - P(A).
• Mutually Exclusive Events: Events that cannot happen at the same time. If A and B are mutually exclusive, P(A or B) = P(A) + P(B).
• Expected Number of Outcomes: To find how many times an event is expected to happen in 'n' trials, calculate P(event) * n.

## Data Types and Collection

• Qualitative Data: Describes qualities or characteristics (e.g., favourite colour).
• Quantitative Data: Deals with numbers.
• Discrete Data: Can only take specific, separate values (e.g., number of siblings).
• Continuous Data: Can take any value within a given range (e.g., height, time).
• Primary Data: Collected by you directly (e.g., surveys).
• Secondary Data: Collected by someone else (e.g., data from the internet).
• Sampling: Taking a smaller, representative group from a larger population. A random sample gives every member an equal chance of being selected.

## Presenting and Analysing Data

• Frequency Tables: Organise data by counting how often each value or group of values appears.
• Two-way Tables: Show the relationship between two categorical variables.
• Bar Charts: Used for discrete or qualitative data, bars don't touch.
• Pie Charts: Show proportions of a whole; calculate angle = (frequency / total frequency) * 360°.
• Line Graphs: Show changes over time.
• Stem and Leaf Diagrams: Display quantitative data, separating each value into a 'stem' (e.g., tens digit) and a 'leaf' (e.g., units digit). Useful for ordering data.
• Measures of Central Tendency: These describe the 'average' or typical value.
• Mode: The most frequent value.
• Median: The middle value when data is ordered from smallest to largest. If two middle values, find their average.
• Mean: Sum of all values / Number of values.
• Measures of Spread: These describe how spread out the data is.
• Range: Highest value - Lowest value.
• Scatter Graphs: Plot two variables to show if there's a relationship (correlation).
• Positive Correlation: As one variable increases, the other tends to increase.
• Negative Correlation: As one variable increases, the other tends to decrease.
• No Correlation: No clear relationship.
• A line of best fit can be drawn by eye to show the general trend and make predictions (interpolation within the data range, extrapolation outside the range – less reliable).