Standard Form

Ever wondered how scientists write those massive numbers like the distance to stars? Standard form is your answer - it's a neat way to write very big or very small numbers without loads of zeros cluttering up your page.

The format is always A × 10ⁿ, where A must be between 1 and 10, and n tells you how many places the decimal point moved. For big numbers like 35,600, you move the decimal point left (35,600 becomes 3.56 × 10⁴), so n is positive. For tiny numbers like 0.0000623, you move it right (becomes 6.23 × 10⁻⁵), so n is negative.

Here's the golden rule: if your original number is bigger than 10, your power is positive. If it's smaller than 1, your power is negative. Once you get this, you'll never mix them up again!

Quick Check: 146.3 million = 146,300,000 = 1.463 × 10⁸ (moved decimal 8 places left)

Standard Form Calculations

Calculating with standard form might look scary, but it's actually easier than working with all those zeros. The trick is to handle the front numbers and powers of 10 separately, then combine them at the end.

For multiplication and division: group the front numbers together and the powers separately. Multiply 2 × 6.75 = 13.5, then use index laws for 10³ × 10⁵ = 10⁸. Your answer becomes 13.5 × 10⁸, but since 13.5 isn't between 1 and 10, convert it to 1.35 × 10⁹.

For addition and subtraction: make sure both numbers have the same power of 10 first. Convert 6.6 × 10³ to 0.66 × 10⁴, then add the front numbers: 9.8 + 0.66 = 10.46 × 10⁴ = 1.046 × 10⁵.

Pro Tip: Always check your final answer is in proper standard form - front number between 1 and 10!

Powers and Roots

Index laws are the backbone of so many maths topics - master these rules and you'll breeze through algebra, standard form, and loads more. Think of them as the grammar rules of mathematical language.

The key rules you absolutely need to know: when multiplying powers, add the indices $x² × x³ = x⁵$. When dividing, subtract them $x⁶ ÷ x² = x⁴$. Anything to the power 0 equals 1, and negative powers mean "flip it" - so 7⁻² = 1/7².

Fractional powers are just roots in disguise. The denominator tells you the type of root: x^(1/2) means square root, x^(1/3) means cube root. For something like 64^(3/4), do the root first $fourth root of 64 = 2$, then the power (2³ = 8).

Memory Trick: Negative indices = "flip it", fractional indices = "root it", then power it!

Circle Theorems

Circle theorems are like a secret code - once you know the rules, you can work out any angle in a circle. These theorems pop up everywhere in GCSE questions, so learning them properly now will save you loads of time later.

The angle at centre is always twice the angle at the circumference when they're looking at the same arc. Any triangle drawn from the ends of a diameter creates a 90° angle where it touches the circumference - this one's incredibly useful for proofs.

Tangents are special because they always meet the radius at 90°, and two tangents from the same external point are always equal length. The alternate segment theorem says the angle between a tangent and chord equals the angle in the opposite segment.

In cyclic quadrilaterals $four-sided shapes in circles$, opposite angles always add up to 180°. Remember these patterns and you'll spot the right theorem to use in any circle problem.

Exam Tip: Draw clear diagrams and mark equal angles with the same symbols - examiners love to see your working!

Pythagoras' Theorem

Pythagoras' theorem is probably the most famous equation in maths: a² + b² = c². It only works for right-angled triangles, where c is always the longest side (the hypotenuse) opposite the right angle.

You'll use this constantly for finding distances, especially between coordinates on graphs. Draw a right-angled triangle connecting your two points, work out the horizontal and vertical distances by subtracting coordinates, then use Pythagoras to find the straight-line distance.

When questions ask for an "exact length," leave your answer as a surd (like 3√3) rather than a decimal. This shows you understand the precise mathematical relationship, not just an approximation.

For coordinate problems, always sketch the triangle first - it stops you making silly mistakes with which sides are which, and helps you visualise what you're actually calculating.

Golden Rule: Always identify which side is the hypotenuse before you start calculating - it's the longest side opposite the right angle!

Basic Trigonometry

Trigonometry connects angles to side lengths in right-angled triangles. The magic words are SOH CAH TOA: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent.

First, identify which angle you're working with, then label the sides: hypotenuse (longest side), opposite (across from your angle), and adjacent (next to your angle). Choose the ratio that includes the two sides you know or need to find.

To find angles, use the inverse functions (sin⁻¹, cos⁻¹, tan⁻¹) on your calculator. These are usually accessed by pressing shift before the trig button. Always check your calculator is in degrees mode, not radians!

Learn the common values for 30°, 45°, and 60° - they appear frequently in non-calculator questions. At 45°, sin and cos both equal 1/√2, which makes sense because you're dealing with an isosceles right triangle.

Calculator Check: Make sure you're in degrees mode - radians will give you completely wrong answers!

Sine and Cosine Rules

When triangles aren't right-angled, sine and cosine rules become your best friends. The sine rule connects angles to opposite sides: a/sin A = b/sin B = c/sin C. The cosine rule is like Pythagoras with an extra bit: a² = b² + c² - 2bc cos A.

Use the sine rule when you know two angles and any side, or two sides and an angle that's not between them. Use the cosine rule when you know two sides and the angle between them, or all three sides but no angles.

For area calculations, use ½ab sin C when you know two sides and the included angle. This is much easier than trying to find the height of the triangle the traditional way.

The cosine rule rearranged $cos A = (b² + c² - a²)/2bc$ is perfect for finding angles when you know all three sides. Don't forget to use cos⁻¹ to get the actual angle from the ratio.

Strategy Tip: Always identify what information you have first - this tells you exactly which rule to use!

Circles and Areas

Circle calculations use π constantly, so get comfortable with it! The basic formulas are Area = πr² and Circumference = 2πr. From these, you can work backwards to find radius or diameter if needed.

Arc length is just a fraction of the full circumference: $angle/360°$ × 2πr. Sector area works the same way: $angle/360°$ × πr². Think of these as "slices of pie" - bigger angles give bigger slices.

For segments (the leftover bit when you remove a triangle from a sector), calculate the sector area and subtract the triangle area using ½r² sin x. This comes up in lots of problem-solving questions.

Different shapes need different area formulas: triangles use ½bh or ½ab sin C, parallelograms use base × height, and trapeziums use ½$a+b$ × h. Surface area formulas for 3D shapes are just combinations of these 2D areas - cylinders are two circles plus a rectangle wrapped around.

Memory Aid: Sectors and arcs both use the same fraction: angle over 360!

Volume Calculations

Volume measures how much space a 3D shape takes up. For prisms $shapes with the same cross-section throughout$, just multiply the cross-sectional area by the length - dead simple!

Curved shapes need special formulas: spheres use (4/3)πr³, cones and pyramids use (1/3) × base area × height. The "one-third" rule for pointy shapes is crucial - don't forget it!

Frustums $cut-off cones$ need a two-step approach: find the volume of the original complete cone, then subtract the volume of the removed top part. Use similar triangles to work out the radius of the removed section.

Always watch your units carefully - volumes are in cubic units (cm³, m³), and remember that 1 litre = 1000 cm³. This trips up loads of students in rate problems where water flows at different units.

Units Warning: Check if you need to convert between litres and cm³ - exam questions love to mix units up!

Rates of Flow

Rate problems combine volume calculations with time, so you need to be sharp with unit conversions. Always convert everything to the same units before calculating - mixing litres and cm³ will mess up your answer completely.

The key is breaking the problem into steps: first find the volume you need to fill, then work out how long it takes at the given rate. For the fish tank example, calculate (2/3) of the sphere's volume, convert the flow rate to cm³/s, then divide volume by rate.

Remember that multiples are a number's times tables (including the number itself), while factors are all numbers that divide into it evenly. These definitions pop up in various contexts throughout your maths course.

Converting between units systematically prevents mistakes: 4 litres/minute = 4000 cm³/minute = 4000÷60 cm³/second. Take it step by step and double-check your conversions.

Time Saver: Always convert to the units the question wants at the start - don't leave it until the end!