# Maths Angel

> AI-powered online maths tutoring platform for students aged 10-16, covering KS3, KS4, and GCSE/IGCSE curricula. Built by EdutopiaTech Ltd in London.

Maths Angel provides video lessons, interactive quizzes, flashcards, and an AI maths assistant called ChatCat. Students can ask any maths question and receive step-by-step help. The platform offers a free tier with 30 lessons and a Pro subscription starting at £9/month for unlimited access.

## Main Pages

- [Home](https://maths-angel.com/): Overview of the platform, featured lessons, and signup
- [Courses](https://maths-angel.com/courses): Full course catalog with KS3, KS4 and GCSE/IGCSE maths coverage
- [ChatCat](https://maths-angel.com/chatcat): AI maths solver with photo, voice, and text input for step-by-step solutions
- [Pricing](https://maths-angel.com/pricing): Free and Pro subscription plans from £9/month
- [About Us](https://maths-angel.com/about-us): Company story, team, and mission

## Number

- [Number Line and Comparing Numbers](https://maths-angel.com/lessons/number-line-inequality-symbols)
  > A number line places numbers in order on a straight line. Learn to compare numbers using inequality symbols like > and < with clear examples. Watch free!
  ### 🛎️ Understanding the Number Line
  - Numbers get **bigger** as you move **right** on the number line.
  - Numbers are spaced at **equal intervals** along the line.
  - ![Number line showing positive and negative numbers, with labels for origin, equal intervals, and tips on comparing numbers using the number line.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/number-line-inequality-symbols/1%29%20Number%20Line.webp)
  ### 🛎️ Using Inequality Symbols
  - The symbol **>** means **greater than** and **<** means **less than**.
  - The symbols **≥** and **≤** include **equal to** as well.
  - ![Inequality symbols explained with examples using the greater than, less than, greater than or equal to, and less than or equal to signs.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/number-line-inequality-symbols/2%29%20Inequality%20Symbol.webp)
  ### 🛎️ Comparing Positive Numbers
  - A number with **more digits** is usually **greater** if both are positive.
  - If the digits are equal, compare from **left to right**. The number with the **first bigger digit** is larger.
  - ![Two rules for comparing numbers: First, more digits mean greater value. Second, if the number of digits is equal, compare from left to right.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/number-line-inequality-symbols/3%29%20Comparing%20Numbers.webp)
- [Place Value and Rounding Numbers](https://maths-angel.com/lessons/place-value-chart-rounding)
  > Place value shows the value of each digit in a number. Learn rounding rules and use place value charts with step-by-step examples. Watch free!
  ### 🛎️ What is a Place Value Chart?
  - A place value chart shows the **value of each digit** in a number.
  - For example, the digit 2 means **20** in tens but **200** in hundreds.
  - ![Place value chart demonstrating the number 54321 with a breakdown of digits in ten thousands, thousands, hundreds, tens, and ones columns.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/place-value-chart-rounding/1%29%20Place%20Value%20Chart.webp)
  ### 🛎️ The Base 10 Number System
  - The base 10 system uses the digits **0-9** to make all numbers.
  - Moving a digit left makes the value **10 times bigger** (5 → 50 → 500).
  - ![Place value chart for the Base 10 number system, also called decimal system, with columns for trillions, billions, millions, thousands, H, T, and O.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/place-value-chart-rounding/2%29%20Base-10%20Number%20System.webp)
  ### 🛎️ How to Round a Number?
  - Find the **target place** you are rounding to, then look at the digit **to the right.**
  - If the digit is **5 or more, round up;** if it is **4 or less, round down.**
  - ![Place value chart showing how to round 5,024,579 to the nearest thousand using digit rules 0-4 down and 5-9 up.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/place-value-chart-rounding/3%29%20Rounding%20Rules%20and%20Steps.webp)
  ### 🛎️ What Does Rounding Down Mean?
  - When rounding down, the target digit **stays the same** and the rest become zeros.
  - For example, **342 rounded to the nearest 10** becomes **340**.
  - ![Place value chart showing how to round down the number 50,245,798 to 50 million by replacing all digits after the thousand place with zeros.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/place-value-chart-rounding/4%29%20Rounding%20down.webp)
  ### 🛎️ What Does Rounding Up Mean?
  - When rounding up, the target digit **increases by 1** and the rest become zeros.
  - For example, **685 rounded to the nearest 10** becomes **690**.
  - ![Place value chart showing how to round up the number 50,245,798 to 50,246,000.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/place-value-chart-rounding/5%29%20Rounding%20up.webp)
- [Natural Numbers, Whole Numbers, and Integers](https://maths-angel.com/lessons/natural-numbers-whole-numbers-integers)
  > Natural numbers start at 1, whole numbers include 0, and integers add negatives. Learn how these number sets relate with examples and practice. Watch free!
  ### 🛎️ Types of Numbers
  - Numbers can be grouped into different **sets**.
  - Common sets are **natural numbers**, **whole numbers**, and **integers**.
  - ![Integers, whole numbers, and natural numbers with examples, notes that natural numbers are whole numbers and whole numbers are integers.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/natural-numbers-whole-numbers-integers/1%29%20Types%20of%20Numbers.webp)
  ### 🛎️ What are Natural Numbers?
  - **Natural numbers** are used for **counting** and **start at 1**.
  - They include **1, 2, 3, 4, 5 …** only.
  - ![Natural numbers 1-5 shown by counting bees, with note that natural numbers exclude zero, negatives, fractions, and decimals](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/natural-numbers-whole-numbers-integers/2%29%20Definition%20of%20Natural%20Numbers.webp)
  ### 🛎️ How to Identify Natural Numbers?
  - Only **positive counting numbers** are **natural numbers**.
  - **0**, **−5**, **1/3**, and **2.5** are **not natural numbers**.
  - ![Practice questions showing zero, −5, 1/3, and 2.5 are not natural numbers](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/natural-numbers-whole-numbers-integers/3%29%20Natural%20Numbers%20Practice.webp)
  ### 🛎️ What are Whole Numbers?
  - **Whole numbers** are **natural numbers plus zero**.
  - They do **not include negative numbers**, **fractions**, or **decimals**.
  - ![Whole numbers include natural numbers and zero, shown as 0, 1, 2, 3, 4, … with note that a whole-number answer can be 0](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/natural-numbers-whole-numbers-integers/4%29%20Definition%20of%20Whole%20Numbers.webp)
  ### 🛎️ How to Identify Whole Numbers?
  - **Zero** is a **whole number**, but not a **natural number**.
  - **Negative numbers** are **not whole numbers**.
  - ![Identifying whole numbers: 0, 17, and 31 are whole numbers but −3, −8, −1.5, 0.3, and ⅖ are not](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/natural-numbers-whole-numbers-integers/5%29%20Whole%20Numbers%20Practice.webp)
  ### 🛎️ What are Integers?
  - **Integers** are whole numbers, their **negatives**, and **zero**.
  - They do **not include fractions or decimals**.
  - ![Integers shown on a keypad as 2, 1, 0, −1, −2, −3, highlighting that integers include whole numbers, their negatives, and zero](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/natural-numbers-whole-numbers-integers/6%29%20Definition%20of%20Integers.webp)
  ### 🛎️ How Are Natural Numbers, Whole Numbers, and Integers Related?
  - Every **natural number** is a **whole number**.
  - Every **whole number** is an **integer**.
  - ![Classifying 5, 0, and −7 as natural numbers, whole numbers, or integers: 5 is all three, 0 is whole and integer, −7 is integer only](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/natural-numbers-whole-numbers-integers/7%29%20Integers%20Practice.webp)
- [Rational Numbers and Their Location on a Number Line](https://maths-angel.com/lessons/rational-numbers)
  > Rational numbers are fractions of two integers. Learn to tell rational from irrational numbers and place them on a number line. Watch free!
  ### 🛎️ What are Rational Numbers?
  - **Rational numbers** can be written as a **fraction** of two **integers**.
  - They include **fractions**, **terminating decimals**, **integers**, and **recurring decimals**.
  - ![Rational numbers explained with examples showing that fractions, integers, terminating decimals, and recurring decimals are rational numbers.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/rational-numbers/1%29%20What%20Are%20Rational%20Numbers.webp)
  ### 🛎️ What are Irrational Numbers?
  - **Irrational numbers** are **decimals** that **never end** and **do not repeat**.
  - They **cannot** be written as a **fraction**.
  - ![Explaining that irrational numbers are non-terminating decimals without repeating patterns, with examples like π and √2.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/rational-numbers/2%29%20What%20Are%20Irrational%20Numbers.webp)
  ### 🛎️ How to Locate Fractions on a Number Line?
  - Rewrite the fraction as a **mixed number**, for example 11/3 = 3+2/3.
  - Go to **3** on the number line, split the gap from **3 to 4** into **3 equal parts**, and move **2 parts** to find **11/3**.
  - ![Locating the rational number and fraction 11/3 on a number line by dividing it into 3 and 2/3, demonstrating on number line.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/rational-numbers/3%29%20Locating%20Fractions%20on%20a%20Number%20Line.webp)
  ### 🛎️ How to Locate Decimals on a Number Line?
  - Rewrite the decimal to separate the whole and decimal parts, for example −2.7 = −2 − 0.7.
  - Go to **−2**, divide the gap from **−2 to −3** into **10 equal parts**, and move **7 parts** to find **−2.7**.
  - ![Locating the rational number and decimal -2.7 on a number line, shown as -2 minus 0.7.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/rational-numbers/4%29%20Rational%20Decimals%20on%20a%20Number%20Line.webp)
- [Absolute Value](https://maths-angel.com/lessons/absolute-value)
  > Absolute value measures distance from zero on a number line. Learn to simplify expressions and work with negative numbers step by step. Watch free!
  ### 🛎️ What is Absolute Value?
  - **Absolute value** is the **distance from zero** on the number line.
  - Absolute value is **never negative**, even for **negative numbers**.
  - ![The definition of absolute value, examples include |5| = 5, |0| = 0, and |-3| = 3.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/absolute-value/1%29%20What%20Is%20Absolute%20Value.webp)
  ### 🛎️ How to Simplify Absolute Values?
  - Always calculate **inside the bars first**.
  - Then apply absolute value to get a **non-negative answer**.
  - ![Two examples simplifying absolute values, showing final results of 9 and 15 after applying absolute value rules.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/absolute-value/2%29%20Simplifying%20Absolute%20Values.webp)
  ### 🛎️ How to Add Negatives with the Same Signs?
  - When both numbers are **negative**, add their **absolute values**.
  - Keep the **negative sign** in the final answer.
  - ![Absolute value calculation when both numbers are negative, e.g., -30 - 120 = -150, by adding absolute values and keeping the negative sign.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/absolute-value/3%29%20Calculating%20Negative%20Numbers%20Case%201.webp)
  ### 🛎️ How to Add Negatives with Different Signs?
  - Subtract the **smaller absolute value** from the **larger**.
  - Keep the **sign of the larger absolute value**.
  - ![Absolute value calculation with different signs by subtracting absolute values and keeping the sign of the larger; e.g., -65 + 15 = -50.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/absolute-value/4%29%20Calculating%20Negative%20Numbers%20Case%202.webp)
- [Basic Arithmetic and Inverse Operations](https://maths-angel.com/lessons/basic-arithmetic-inverse-operations)
  > Basic arithmetic covers addition, subtraction, multiplication, and division. Learn how inverse operations undo each other with clear examples. Watch free!
  ### 🛎️ What is Addition?
  - Addition is the process of **combining numbers** to find a total.
  - For example, **7 + 5 = 12**, where the answer is called the **sum.**
  - ![Visual explanation of addition 6 + 6 + 6 = 18, highlighting summands and the sum.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/basic-arithmetic-inverse-operations/1%29%20Addition.webp)
  ### 🛎️ What is Multiplication?
  - Multiplication is **repeated addition** of the same number.
  - For example, **3 × 6 = 18** means adding **6 three times** (6 + 6 + 6).
  - ![Explanation of multiplication 3 x 6 = 18, showing multiplier, multiplicand, and product. ​](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/basic-arithmetic-inverse-operations/2%29%20Multiplication.webp)
  ### 🛎️ What is Subtraction?
  - Subtraction is the process of **taking away** one number from another.
  - For example, **18 − 6 = 12**, where the answer is called the **difference.**
  - ![Subtraction explained with an example showing 18 minus 6 equals 12, including terms minuend, subtrahend, and difference.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/basic-arithmetic-inverse-operations/3%29%20Subtraction.webp)
  ### 🛎️ What is Division?
  - Division means **sharing a total equally** into groups.
  - For example, **12 ÷ 4 = 3**, imagine sharing 12 apples equally into 4 groups, each group has 3 apples.
  - ![Division of 12 by 4 is explained, including the terms dividend, divisor, and quotient, and an explanation of why division by zero is impossible.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/basic-arithmetic-inverse-operations/4%29%20Division.webp)
  ### 🛎️ What are Inverse Operations?
  - Inverse operations **undo each other.**
  - Addition undoes subtraction, and multiplication undoes division.
  - ![Diagram illustrating inverse operations in maths, showing examples of addition, subtraction, multiplication, and division on number lines.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/basic-arithmetic-inverse-operations/5%29%20Inverse%20Operations.webp)
- [BIDMAS - Order of Operations](https://maths-angel.com/lessons/bidmas-order-of-operations)
  > BIDMAS explains the order of operations: Brackets, Indices, Division, Multiplication, Addition, Subtraction. Practice with worked examples. Watch free!
  ### 🛎️ What is a Mathematical Expression?
  - A mathematical expression is a calculation made using **numbers and operations** (+, −, ×, ÷).
  - For example, **3 + 4 × 2** is an expression, but **3 + 4 = 7** is an equation.
  - ![Total cost of a £300 phone and £20 each for a charger, headphones, and mouse, paid in 12 installments, as an expression using terms and operators.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/bidmas-order-of-operations/1%29%20Mathematical%20Expression.webp)
  ### 🛎️ What is BIDMAS?
  - **BIDMAS** tells you the correct order to calculate.
  - It stands for **Brackets**, **Indices**, **Division**, **Multiplication**, **Addition**, **Subtraction**.
  - ![Expression (300 + 3 × 20) ÷ 12 with terms and operators labelled, applying BIDMAS to a real-world shopping problem.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/bidmas-order-of-operations/2%29%20BIDMAS.webp)
  ### 🛎️ How to Apply BIDMAS?
  - If there are **brackets**, calculate what is **inside them** first.
  - If two operations are at the **same level**, work from **left to right**.
  - ![Practising BIDMAS with examples such as of division and multiplication, and addition and subtraction.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/bidmas-order-of-operations/3%29%20Applying%20BIDMAS.webp)
  ### 🛎️ How to Solve Multiple Brackets?
  - If there is more than one set of brackets, start with the **innermost brackets**.
  - Then continue working **outwards**, following **BIDMAS**.
  - ![Step by step example of solving an expression with multiple brackets, 50-(2*(2+4*5)-3), using BIDMAS/BODMAS.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/bidmas-order-of-operations/4%29%20Dealing%20with%20Multiple%20Brackets.webp)
- [Commutative and Associative Properties](https://maths-angel.com/lessons/commutative-property-associative-property)
  > The commutative property lets you swap numbers, and the associative property lets you regroup them. Works for addition and multiplication. Watch free!
  ### 🛎️ What is the Commutative Property?
  - The commutative property means you can **swap the order of** numbers
  - It works for **addition and multiplication,** but **not** for subtraction or division
  - ![Commutative property showing number swapping in addition and multiplication, not applicable to subtraction or division](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/commutative-property-associative-property/1%29%20The%20Commutative%20Law.webp)
  ### 🛎️ Commutative Property: Examples
  -	In addition, changing the order gives the **same total**
  - In multiplication, changing the order gives the **same product**
  - ![Commutative law examples in addition (77 + 246 + 23) and multiplication (25 × 19 × 4) with step-by-step solutions](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/commutative-property-associative-property/2%29%20Commutative%20Law%20Example.webp)
  ### 🛎️ What is the Associative Property?
  - The associative property means you can **change how numbers are grouped**
  - It works for **addition and multiplication,** but **not** for subtraction or division
  - ![The Associative Law allows grouping numbers when adding or multiplying, but does not apply to subtraction or division.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/commutative-property-associative-property/3%29%20The%20Associative%20Law.webp)
  ### 🛎️ Associative Property: Examples
  - In addition, regrouping numbers does **not change the total**
  - In multiplication, regrouping numbers does **not change the product**
  - ![Associative law examples in addition (127 + 48) + 52 and multiplication 5 × (2 × 228) with worked solutions](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/commutative-property-associative-property/4%29%20Associative%20Law%20Example.webp)
  ### 🛎️ Using Both Properties Together
  - You can **reorder and regroup** numbers to make calculations easier
  - This helps you with mental maths and **simplifying** expressions
  - ![Combining commutative and associative laws in addition 32 + (115 + 68) and multiplication 20 × (48 × 5)](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/commutative-property-associative-property/5%29%20Combining%20Commutative%20and%20Associative%20Law.webp)
- [Column Addition and Column Subtraction](https://maths-angel.com/lessons/column-addition-and-subtraction)
  > Column addition and subtraction are used to add and subtract multi-digit numbers vertically. Learn carrying and borrowing with clear examples. Watch free!
  ### 🛎️ How to Do Column Addition
  - Line up the digits **vertically** and start adding **from the right**
  - If a column totals **10 or more, carry 1** to the next column on the left
  - ![Column addition for 3425 + 782, showing how to align the numbers vertically. Start addition from the right, and carry over for sums over ten.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/column-addition-and-subtraction/1%29%20Column%20Addition%20and%20Carrying%20Over.webp)
  ### 🛎️ How to Do Column Subtraction
  - Write the **larger number on top** and subtract **from the right**
  - If the top digit is smaller, **borrow 1** from the next column on the left
  - ![Column subtraction example showing how to subtract 876 from 2826, with steps including borrowing and aligning numbers, resulting in 1950.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/column-addition-and-subtraction/2%29%20Column%20Subtraction%20and%20Borrowing.webp)
- [Long Multiplication](https://maths-angel.com/lessons/long-multiplication)
  > Long multiplication splits large calculations into smaller digit-by-digit steps. Learn to multiply 2-digit and 3-digit numbers with clear examples. Watch free!
  ### 🛎️ What is Long Multiplication?
  - Long multiplication splits a **large multiplication** into **smaller digit-by-digit steps.**
  - You multiply each digit, then **add the results together** to get the final answer.
  - ![Long multiplication of 352 by 24 showing step-by-step partial products and final answer 8448.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/long-multiplication/1%29%20Long%20Multiplication%20Example%201.webp)
  ### 🛎️ Key Reminders to Avoid Mistakes
  - Each new row starts **one place further left** because the value is **10 times bigger.**
  - Always check **column alignment.** Misaligned columns give the wrong answer.
  - ![Long multiplication of 67 by 52 broken into steps, showing partial products and final answer 3484.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/long-multiplication/2%29%20Long%20Multiplication%20Example%202.webp)
- [Long Division](https://maths-angel.com/lessons/long-division)
  > Long division divides large numbers into smaller steps. Learn dividend, divisor, quotient and remainder with step-by-step worked examples. Watch free!
  ### 🛎️ Understanding the Parts of Long Division
  - The **dividend** is the number being divided, and the **divisor** is the number you divide by.
  - The **quotient** is the answer on top, and the **remainder** is what is left at the end.
  - ![Long division example solving 364 divided by 30, showing steps with quotient 12 and remainder 4, including a check with reverse calculation.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/long-division/1%29%20Long%20Division%20Example%201.webp)
  ### 🛎️ Checking a Long Division Answer
  - You can check your answer using: **Quotient × Divisor + Remainder = Dividend.**
  - For example, **230 × 7 + 5 = 1615**, so **1615 ÷ 7 = 230 r 5** is correct.
  - ![Long division example of 1615 divided by 7, showing step-by-step process with a final result of 230 remainder 5.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/long-division/2%29%20Long%20Division%20Example%202.webp)
- [Movement of the Decimal Point](https://maths-angel.com/lessons/decimal-point-movement)
  > Moving the decimal point right multiplies a number by 10, 100, or 1000. Moving it left divides. Learn the rules with clear examples. Watch free!
  ### 🛎️ Multiplying and Dividing by 10, 100, 1000
  - **Multiplying** by 10, 100, 1000 moves the decimal **right**.
  - **Dividing** by 10, 100, 1000 moves the decimal **left**.
  - ![Shifting the decimal point in division and multiplication, showing how the decimal point shifts left for division and right for multiplication.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/decimal-point-movement/1%29%20Movement%20of%20the%20Decimal%20Point.webp)
  ### 🛎️ How to Shift the Decimal Point
  - **Multiplying by 1000** means moving the decimal point **3 places to the right**.
  - **Dividing by 100** means moving the decimal point **2 places to the left**.
  - ![Examples on how to move the decimal point when calculating 5.02*1000, 4.21/100, and 56.7%*10.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/decimal-point-movement/2%29%20Movement%20of%20the%20Decimal%20Point%20Examples.webp)
  ### 🛎️ Using Decimal Movement to Convert Units
  - When converting to **smaller units,** **multiply** by the conversion factor. For example, 2.8m = 2.8 × 100 = 280cm
  - When converting to **larger units,** **divide** by the conversion factor. For example, 2.8m = 2.8 ÷ 1000 = 0.0028km
  - ![Converting 2.8 metres to 280 centimetres by multiplying by 100, and converting 0.0028 kilometres to metres by multiplying by 1000.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/decimal-point-movement/3%29%20Movement%20of%20the%20Decimal%20Point%20in%20Unit%20Conversion.webp)
- [Adding and Subtracting Negative Numbers](https://maths-angel.com/lessons/adding-and-subtracting-negative-numbers)
  > Learn the rules for adding and subtracting negative numbers: same signs become plus, different signs become minus. Practice with examples. Watch free!
  ### 🛎️ What Are Negative Numbers?
  - **Negative numbers** are to the **left of zero** on a number line.
  - **Zero** is neither positive nor negative.
  - ![Number line showing negative and positive numbers from -5 to 5, with zero as neutral, labelled 'Understanding Negative Numbers'.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/adding-and-subtracting-negative-numbers/1%29%20Understanding%20Negative%20Numbers.webp)
  ### 🛎️ How to Add and Subtract Using a Number Line?
  - **Adding** a positive number means **move right** on the number line.
  - **Subtracting** a positive number means **move left** on the number line.
  - ![Number line visual for adding and subtracting negative numbers, showing (-7) + 5 = -2 with a move right and (-6) - 3 = -9 with a move left. ​](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/adding-and-subtracting-negative-numbers/2%29%20Adding%20and%20Subtracting%20Negative%20Numbers%20Using%20Number%20Line.webp)
  ### 🛎️ How Are Negative Numbers Used in Real Life?
  - A **drop** in temperature means you **subtract**.
  - You may end with a **negative answer** if you go below **zero**.
  - ![Temperature drops from 3°C to -7°C by subtracting 10°C, shown on a number line to teach negative numbers.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/adding-and-subtracting-negative-numbers/3%29%20Application%20-%20Negative%20Numbers.webp)
  ### 🛎️ How to Remove Brackets?
  - **Same signs** next to brackets become a **plus**.
  - **Different signs** next to brackets become a **minus**.
  - ![Rules for removing brackets when adding or subtracting negative numbers using same and different sign combinations.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/adding-and-subtracting-negative-numbers/4%29%20Rules%20for%20Removing%20Brackets%20for%20Addition%20and%20Subtraction.webp)
  ### 🛎️ How to Remove Brackets with Several Numbers?
  - If there is a **minus before brackets**, **change all signs inside**.
  - You can also work out **what is inside the brackets first**.
  - ![Steps to remove brackets with multiple numbers, solving -12 - (-12 + 5) by solving inside first or applying the rule to change signs. Result is -5.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/adding-and-subtracting-negative-numbers/5%29%20Removing%20Brackets%20with%20Multiple%20Numbers.webp)
- [Multiplying and Dividing Negative Numbers](https://maths-angel.com/lessons/multiplying-and-dividing-negative-numbers)
  > Multiplying and dividing negative numbers follows sign rules: an odd count of negatives gives a negative, even gives a positive. Learn with examples. Watch free!
  ### 🛎️ Sign Rules for Multiplication and Division
  - An **odd number of negative signs** gives a **negative answer**, for example 5 × (−3) = −15.
  - An **even number of negative signs** gives a **positive answer**, for example (−12) ÷ (−3) = +4.
  - ![Multiplication and division of negative numbers, showing odd negative signs give a negative result and even negative signs give a positive result.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/multiplying-and-dividing-negative-numbers/1%29%20Sign%20Rules%20of%20Multiplication%20and%20Division.webp)
  ### 🛎️ Multiplying and Dividing Three Numbers
  - You can **work out two numbers at a time** to make it easier.
  - Count the **negative signs** to double-check the **final sign**.
  - ![Steps for multiplication and division with rules for odd and even negative signs determining result positivity.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/multiplying-and-dividing-negative-numbers/2%29%20Multiplying%20and%20Dividing%203%20Numbers.webp)
  ### 🛎️ Combining Multiplication and Division
  - Multiply and divide **from left to right**.
  - Count **how many negatives** there are to decide if the answer is **positive or negative**.
  - ![Combining multiplication and division with examples showing the rules for odd and even numbers of negative signs, calculating from left to right.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/multiplying-and-dividing-negative-numbers/3%29%20Combining%20Multiplication%20and%20Division.webp)
- [Divisibility and Divisibility Rules](https://maths-angel.com/lessons/divisibility-rules)
  > Divisibility rules for 2 to 10 are simple digit checks to test if a number divides evenly. Learn each rule with clear worked examples. Start free today!
  ### 🛎️ What Does Divisible Mean?
  - A number is **divisible** if it divides **exactly with** no remainder.
  - For example, **12 ÷ 3 = 4** (divisible), but **13 ÷ 3 = 4 r 1** (not divisible).
  - ![Divisibility explained with bananas — dividing 9 by 2 leaves a remainder, but dividing by 3 gives no remainder](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/divisibility-rules/1%29%20Divisibility.webp)
  ### 🛎️ Divisibility Rules for 2, 4, and 8
  - A number is divisible by **2** if the **last digit is even** (0, 2, 4, 6, 8).
  - A number is divisible by **4** if the **last two digits** are divisible by 4.
  - A number is divisible by **8** if the **last three digits** are divisible by 8.
  - ![Divisibility rules for 2, 4, and 8 based on last digits with examples 26, 328, and 6160](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/divisibility-rules/2%29%20Divisibility%20Rules%20for%202%2C4%2C8.webp)
  ### 🛎️ Divisibility Rules for 3, 6, and 9
  - A number is divisible by 3 if the **sum of its digits** is divisible by 3.
  - A number is divisible by 6 if it is **divisible by both 2 and 3**.
  - A number is divisible by 9 if the **sum of its digits** is divisible by 9.
  - ![Divisibility rules for 3, 6, and 9 using digit sum method with examples 543, 198, and 402](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/divisibility-rules/3%29%20Divisibility%20Rules%20for%203%2C6%2C9.webp)
  ### 🛎️ Divisibility Rules for 5 and 10
  - A number is divisible by 5 if the last digit is **0 or 5**.
  - A number is divisible by 10 if the last digit is **0**.
  - ![Divisibility rules for 5 and 10 — last digit must be 0 or 5 for five, and 0 for ten, with examples](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/divisibility-rules/4%29%20Divisibility%20Rules%20for%205%2C%2010.webp)
- [Prime Numbers and Prime Factorisation](https://maths-angel.com/lessons/prime-numbers-prime-factorisation)
  > A prime number has exactly two factors: 1 and itself. Learn prime factorisation using factor trees and division with step-by-step examples. Watch free!
  ### 🛎️ What is a Prime Number?
  - A prime number has **exactly two different factors:** 1 and itself.
  - **2** is the smallest prime number.
  - 0 and 1 are **not prime numbers**.
  - ![Explanation and example of prime numbers, stating that 2 is the smallest prime number.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/prime-numbers-prime-factorisation/1%29%20Prime%20Numbers.webp)
  ### 🛎️ Prime Numbers Below 20
  - The prime numbers below 20 are: **2, 3, 5, 7, 11, 13, 17, 19.**
  - Remembering them helps you factorise larger numbers.
  - ![Cartoon dolphin showing prime numbers under 20, which are 2, 3, 5, 7, 11, 13, 17, and 19.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/prime-numbers-prime-factorisation/2%29%20Prime%20Numbers%20below%2020.webp)
  ### 🛎️ What Does Prime Factorisation Mean?
  - Prime factorisation means writing a number as a **product of prime numbers only.**
  - For example, 90 = 2 × 3 × 3 × 5.
  - ![Prime factorisation of 90 using the division method step by step.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/prime-numbers-prime-factorisation/3%29%20Prime%20Factorisation%20Division%20Method.webp)
  ### 🛎️ Factor Tree Method for Prime Factorisation
  - The factor tree method breaks a number into smaller factors step by step.
  - Keep factorising **until all factors are prime numbers.**
  - ![Prime factorisation of 126 using the factor tree method step by step.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/prime-numbers-prime-factorisation/4%29%20Prime%20Factorisation%20Tree%20Method.webp)
- [How to Find HCF and LCM](https://maths-angel.com/lessons/hcf-and-lcm)
  > The HCF is the largest factor two numbers share, and the LCM is their smallest common multiple. Master both methods with step-by-step examples. Watch free!
  ### 🛎️ What Are HCF and LCM?
  - **HCF** (Highest Common Factor) is the **biggest factor** the numbers share
  - **LCM** (Lowest Common Multiple) is the **smallest multiple** the numbers share
  - ![Explanation that HCF is the biggest factor shared and LCM is the smallest multiple shared.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/hcf-and-lcm/1%29%20What%20is%20HCF%20and%20LCM.webp)
  ### 🛎️ Finding HCF and LCM by Listing
  -	For **HCF,** list the **factors** of each number and choose the **largest common one**
  - For **LCM,** list the **multiples** of each number and choose the **smallest common one**
  - ![Method for finding Highest Common Factor (HCF) and Lowest Common Multiple (LCM) by listing factors and multiples, with HCF of 6 and LCM of 36.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/hcf-and-lcm/2%29%20Finding%20HCF%20and%20LCM%20by%20Listing.webp)
  ### 🛎️ Finding HCF and LCM by Prime Factorisation
  - Use **prime factorisation** for larger numbers or when listing would be long
  - Write each number as a product of **prime factors**
  - ![Using prime factorisation to find HCF and LCM for numbers 180 and 168, with step-by-step calculations and final results.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/hcf-and-lcm/3%29%20Finding%20HCF%20and%20LCM%20by%20Prime%20Factorisation.webp)
  ### 🛎️ Using Prime Factors to Find HCF and LCM
  - For **HCF,** take the **common primes** with the **lowest powers**
  - For **LCM,** take **all primes** with the **highest powers**
  - ![Prime factorisation of 56 and 100 to find HCF as 2² = 4 and LCM as 2³ × 5² × 7 = 1400.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/hcf-and-lcm/4%29%20Finding%20HCF%20and%20LCM%20-%20Further%20Example.webp)
- [Introduction to Fractions](https://maths-angel.com/lessons/introduction-to-fractions)
  > Fractions represent parts of a whole using a numerator and denominator. Learn what fractions are, find fractions of amounts, and more. Start free today.
  ### 🛎️ What is a Fraction?
  - A fraction shows **part of a whole.**
  - For example, if a pizza is cut into **8 equal slices** and you take **3,** the fraction is **3/8.**
  - ![Explanation of fractions with a pizza divided into eight slices, illustrating one-eighth and three-eighths fractions.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/introduction-to-fractions/1%29%20Introduction%20to%20Fractions.webp)
  ### 🛎️ Parts of a Fraction (Example: 3/8)
  - The **numerator (3)** shows how many parts are taken.
  - The **denominator (8)** shows how many **equal parts** the whole is split into.
  - ![Explanation of fractions with numerators and denominators, showing a pizza divided into eight equal parts.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/introduction-to-fractions/2%29%20What%20is%20a%20Fraction.webp)
  ### 🛎️ Exam Rule: Equal Parts
  - Fractions only work when the whole is divided into **equal parts.**
  - If the parts are **not equal**, you **cannot** write a correct fraction.
  - ![Understanding fractions with examples, showing 4 out of 9 coloured parts in a circle and an unequally divided rectangle.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/introduction-to-fractions/3%29%20Understanding%20Fractions.webp)
  ### 🛎️ How Do You Find a Fraction of an Amount?
  - **Divide** the total by the denominator to find one part: **20 ÷ 4 = 5.**
  - **Multiply** that by the numerator to get the answer: **5 × 3 = 15.**
  - So, **3/4 of 20 = 15.**
  - ![Applying fractions to real-world examples, including finding 3/4 of 20 cm using a ruler and 5/6 of 2 hours using a clock.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/introduction-to-fractions/4%29%20Fractions%20Examples.webp)
  ### 🛎️ Finding Missing Fractions
  - The parts of a whole **must add up to 1.**
  - To find a missing fraction, **subtract the known fractions from 1.**
  - ![Splitting £50 using fractions: the calculation shows it is divided into 10 parts, 1 part = £5, your share = 3 parts, total = £15.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/introduction-to-fractions/5%29%20Applying%20Fractions.webp)
- [Equivalent Fractions](https://maths-angel.com/lessons/equivalent-fractions)
  > Equivalent fractions look different but represent the same value, like 1/2 = 2/4. Learn to expand and simplify fractions with clear examples. Watch free!
  ### 🛎️ Expanding Fractions
  - To expand a fraction, **multiply** the numerator and denominator **by the same number.**
  - This changes how the fraction **looks,** but the **value stays the same.**
  - ![Expanding fractions example showing 1/2 multiplied by 8 to get 8/16 with visual representation of equivalent fractions 1/2, 2/4, 4/8, and 8/16.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/equivalent-fractions/1%29%20Expanding%20Fractions.webp)
  ### 🛎️ Simplifying Fractions
  - To simplify a fraction, **divide** the numerator and denominator **by the same number.**
  - This keeps the **value the same,** but makes the fraction **simpler.**
  - ![Illustration shows simplifying fractions with examples of 8/16 simplified to 1/2 and equivalent fractions 1/2, 2/4, 4/8, and 8/16 using shaded boxes.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/equivalent-fractions/2%29%20Simplifying%20Fractions.webp)
  ### 🛎️ How to Expand Fractions
  - Find the number that both the numerator and denominator are multiplied by.
  - For example, **2 × 9 = 18,** so **7 × 9 = 63.**
  - So, 2/7 can be expanded by 9 to make 18/63.
  - ![Simplifying 18/63 to 2/7 by dividing by 9 and expanding 2/7 to 18/63 by multiplying by 9.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/equivalent-fractions/3%29%20Expanding%20and%20Simplifying%20Fractions%20Examples.webp)
  ### 🛎️ How to Simplify Fractions
  - Find the number that both the numerator and denominator are divided by.
  - For example, **72 ÷ 6 = 12** and **120 ÷ 6 = 20.**
  - So, 72/120 can be simplified by 6 to make 12/20.
  - ![Expanding fractions by multiplying and simplifying fractions by dividing using the example of 12/20 and 72/120.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/equivalent-fractions/4%29%20Expanding%20and%20Simplifying%20Fractions%20Practice.webp)
  ### 🛎️ Fully Simplified Fractions
  - A fraction is fully simplified when the numerator and denominator have **no common factors other than 1.**
  - For example, **6/8** is not fully simplified, but **3/4** is.
  - ![Fully simplified fractions example, showing the fraction 480/1260 simplified to 8/21 by dividing both numerator and denominator by common factors.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/equivalent-fractions/5%29%20Fully%20Simplified%20Fractions.webp)
- [Comparing Fractions](https://maths-angel.com/lessons/comparing-fractions)
  > Comparing fractions shows which is larger or smaller. Learn to compare fractions with the same or different denominators using the common denominator method and clear examples!
  ### 🛎️ Quick Comparison Rules
  - If the **denominators are the same,** the fraction with the **larger numerator** is greater.
  - If the **numerators are the same,** the fraction with the **smaller denominator** is greater.
  - ![Comparing fractions, showing 5/8 is greater than 3/8 with chocolate bars, and 4/5 is greater than 4/9 with pizza slices.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/comparing-fractions/1%29%20Rules%20for%20Comparing%20Fractions.webp)
  ### 🛎️ Comparing Fractions Using a Common Denominator
  - Find the **lowest common multiple (LCM)** of the denominators. For example, the LCM of 6 and 10 is **30**.
  - Convert both fractions to have **this same denominator**, then compare numerators.
  - ![Steps to compare fractions by finding the lowest common multiple (LCM) of denominators and converting fractions to have the same denominator.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/comparing-fractions/2%29%20Common%20Denominator%20Method.webp)
  ### 🛎️ When One Denominator Is Already the LCM
  - Sometimes one fraction **already has the LCM as its denominator.**
  - Only change the **other fraction,** then compare the **numerators.**
  - ![Comparing fractions by finding the lowest common multiple (LCM) of denominators, converting fractions, and identifying the larger numerator.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/comparing-fractions/3%29%20Practise%20Comparing%20Fractions%20part%201.webp)
  ### 🛎️ When Denominators Are Prime to Each Other
  - If the denominators have **no common factors,** the LCM is their **product.**
  - When the denominators are the same, you can ignore them and just **compare the numerators.**
  - ![Step-by-step method comparing 4/11 and 5/12 by finding LCM 132, converting fractions to 48/132 and 55/132, showing 4/11 < 5/12.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/comparing-fractions/4%29%20Practise%20Comparing%20Fractions%20part%202.webp)
- [Adding and Subtracting Fractions](https://maths-angel.com/lessons/adding-and-subtracting-fractions)
  > Adding and subtracting fractions requires a common denominator when fractions are unlike. Learn with proper, improper, and mixed numbers. Start free today!
  ### 🛎️ Adding and Subtracting Like Fractions
  - **Like fractions** have the **same denominator**.
  - Add or subtract the numerators only, keep the denominator the same, then simplify if possible.
  - ![Explanation of the addition of fractions with the same denominator using pizza slices to illustrate.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/adding-and-subtracting-fractions/1%29%20Adding%20Subtracting%20Like%20Fractions.webp)
  ### 🛎️ Adding and Subtracting Unlike Fractions
  -	**Unlike fractions** have **different denominators**.
  -	**Make the denominators the same first,** then add or subtract.
  - ![Adding and subtracting fractions with different denominators. Showing the steps to make the denominators the same and subtract fractions 5/6 and 1/4.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/adding-and-subtracting-fractions/2%29%20Adding%20Subtracting%20Unlike%20Fractions.webp)
  ### 🛎️ Types of Fractions
  - **Proper fractions:** numerator is **smaller than** the denominator (e.g.  \dfrac{1}{3} )
  - **Improper fractions:** numerator is **greater than or equal to** the denominator (e.g.  \dfrac{8}{5} )
  - **Mixed numbers:** a whole number and a fraction together (e.g.  2\dfrac{3}{5} )
  - ![Different types of fractions in a diagram showing proper fractions, improper fractions, and mixed fractions with examples.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/adding-and-subtracting-fractions/3%29%20Proper%2C%20Improper%2C%20Mixed%20Fractions.webp)
  ### 🛎️ Converting Between Improper and Mixed Fractions
  - **Improper → Mixed:** divide the numerator by the denominator.
  - **Mixed → Improper:** multiply the whole number by the denominator, then add the numerator.
  - ![Converting between improper fractions and mixed numbers with step-by-step calculations.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/adding-and-subtracting-fractions/4%29%20Improper%20Fractions%20to%20Mixed%20Fractions.webp)
- [Multiplying Fractions](https://maths-angel.com/lessons/multiplying-fractions)
  > Multiplying fractions means multiplying numerators and denominators, then simplifying. Learn cross-cancelling with whole and mixed numbers. Watch free!
  ### 🛎️ How Do We Multiply Fractions?
  - Multiply the **numerators together** and multiply the **denominators together**.
  - Then **simplify the fraction**, if possible.
  - ![Step-by-step method for multiplying fractions: multiply numerators, multiply denominators, then simplify the result](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/multiplying-fractions/1%29%20Multiplying%20Fractions%20and%20Cross-Cancelling.webp)
  ### 🛎️ What Is Cross-Cancelling When Multiplying Fractions?
  - You can simplify first by **cross-cancelling a numerator with a denominator**.
  - Cross-cancelling makes the numbers **smaller and easier** to multiply.
  - ![Cross-cancelling example showing how to simplify before multiplying fractions for an easier calculation](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/multiplying-fractions/2%29%20Practising%20Cross-Cancelling.webp)
  ### 🛎️ How Do We Multiply a Fraction by a Whole Number?
  - Write the whole number **as a fraction over 1.**
  - Then multiply as you would with two fractions.
  - ![Multiplying a fraction by a whole number: 88 times 13/22 using cross-cancelling to find 52 black piano keys](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/multiplying-fractions/3%29%20Fraction%20times%20Number.webp)
  ### 🛎️ How Do We Multiply Mixed Numbers?
  - Convert mixed numbers **into improper fractions first.**
  - Then multiply and simplify the final answer.
  - ![Converting mixed numbers to improper fractions before multiplying, with worked example and final answer](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/multiplying-fractions/4%29%20Multiplying%20Mixed%20Fractions.webp)
- [Dividing Fractions](https://maths-angel.com/lessons/dividing-fractions)
  > Dividing fractions means keeping the first fraction, flipping the second, and multiplying. Learn to divide by whole and mixed numbers. Start free today.
  ### 🛎️ How to Divide a Fraction?
  - Change the **division** sign into **multiplication**.
  - Turn the **second fraction upside down** to get the **reciprocal**.
  - **Multiply** the fractions and **simplify** to get the final answer.
  - ![Step-by-step dividing fractions method: change division to multiplication, flip the second fraction, and simplify](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/dividing-fractions/1%29%20Division%20with%20Fractions.webp)
  ### 🛎️ How to Divide a Fraction by a Whole Number?
  - Write the **whole number** as a **fraction over 1**.
  - Flip it to its **reciprocal**, then **multiply**.
  - ![Dividing a fraction by a whole number: writing it as a fraction, flipping to the reciprocal, and cross-cancelling](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/dividing-fractions/2%29%20Dividing%20Fractions%20by%20a%20Whole%20Number.webp)
  ### 🛎️ How to Handle Mixed Numbers in Division?
  - First convert **mixed numbers** into **improper fractions**.
  - To do so, multiply the **whole number** by the **denominator**, then **add the numerator**.
  - Finally, use the same rule to divide: flip, multiply, and simplify.
  - ![Dividing mixed numbers: converting 2⅖ to an improper fraction and dividing by ⅖ using the reciprocal method](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/dividing-fractions/3%29%20Handling%20Mixed%20Fractions.webp)
- [Decimals and Fractions](https://maths-angel.com/lessons/decimals-and-fractions)
  > Decimals and fractions are two ways to show parts of a number. Learn place value and how to convert between them with step-by-step examples. Start free today!
  ### 🛎️ The Decimal Place Value
  - The **decimal point** separates the **whole number** and the **fractional part**.
  - Each place to the **right** of the decimal point is **10 times smaller:** tenths, hundredths, thousandths.
  - ![Place value chart introducing decimals, sort digits from hundreds to thousandths, with the number 12.345 split into whole number and fractional parts.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/decimals-and-fractions/1%29%20Decimal%20and%20Decimal%20Point.webp)
  ### 🛎️ Converting Decimals (Less Than 1) to Fractions
  - Write the decimal as a fraction over **10, 100, or 1000.**
  - Simplify the fraction if possible. For example, 0.75 = \dfrac{75}{100} = \dfrac{3}{4}.
  - ![Showing decimal to fraction, 0.1 as 1/10, 0.01 as 1/100, 0.001 as 1/1000, 0.4 as 4/10 or 2/5, 0.75 as 75/100 or 3/4, and 0.028 as 28/1000 or 7/250.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/decimals-and-fractions/2%29%20Converting%20Decimals%20to%20Fractions.webp)
  ### 🛎️ Converting Decimals (Greater Than 1) to Fractions
  - Split the number into a **whole number** and a **decimal part.**
  - Convert the **decimal part to a fraction**, then **combine** with the whole number.
  - For example, 8.25 = 8 + 0.25 = 8 + 1/4 = 81/4
  - ![Decimals greater than 1 to fractions, showing conversion of 8.25 to 825/100, 165/20, and 33/4, and breaking 8.25 into 8 and 0.25 as 8 + 1/4.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/decimals-and-fractions/3%29%20Converting%20Decimals%20greater%20than%201%20to%20Fractions.webp)
- [Comparing and Rounding Decimals](https://maths-angel.com/lessons/comparing-and-rounding-decimals)
  > Learn to compare and round decimals to the nearest tenth or hundredth. Check the digit to the right: 5 or more rounds up, less rounds down. Watch free!
  ### 🛎️ What Is Decimal Place Value?
  - The **1st, 2nd,** and **3rd decimal places** are **tenths, hundredths, thousandths.**
  - For example, 12.348 = 12 + 0.3 + 0.04 + 0.008.
  - ![Decimal places and place value chart showing number 12.348 with illustrations highlighting tenths, hundredths, and thousandths.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/comparing-and-rounding-decimals/1%29%20Decimal%20Places%20and%20Decimal%20Place%20Value.webp)
  ### 🛎️ Rounding Decimals Rules
  - Look at the digit **one place to the right** of the rounding place.
  - If it is **less than 5,** round **down**. If it is **5 or more,** round **up**.
  - For example, 12.348 rounded to the **nearest tenth** is **12.3**, because the digit in the hundredths place is **4**, which is less than 5, so we **round down**.
  - ![Rounding decimals, showing rules for rounding up if the digit is 5 or greater, and rounding down if the digit is less than 5, with examples.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/comparing-and-rounding-decimals/2%29%20Rounding%20Decimals%20Rules.webp)
  ### 🛎️ Comparing Decimals
  - Line up the **decimal points** first.
  - Compare digits from **left to right.**
  - The number with the **larger digit first** is greater.
  - ![Comparison of two decimals, 12.431 and 12.441, with steps to align decimal points, check digits from left to right, and determine the greater decimal.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/comparing-and-rounding-decimals/3%29%20How%20to%20Compare%20Decimals.webp)
- [Convert Fraction to Decimal Using Long Division](https://maths-angel.com/lessons/convert-fraction-to-decimal)
  > A fraction converts to a decimal by dividing the numerator by the denominator. The result is either terminating or recurring. Learn with worked examples. Watch free!
  ### 🛎️ How Do We Convert Simple Fractions to Decimals?
  - Some fractions can be changed into decimals by rewriting them over **10, 100, or 1000.**
  - Example: \dfrac{2}{5} = \dfrac{4}{10} = 0.4
  - ![Converting fractions like 2/5 and 3/4 to decimals using equivalent fractions and prompting 21/16 as a challenge.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/convert-fraction-to-decimal/1%29%20Converting%20Fractions%20to%20Decimals.webp)
  ### 🛎️ Converting Fractions to Decimals Using Long Division
  - Divide the **numerator by the denominator.**
  - Stop dividing when the **remainder becomes 0.**
  - For example, \dfrac{21}{16} = 1.3125
  - ![Convert fractions to decimals using long division with step-by-step example 21 ÷ 16 = 1.3125](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/convert-fraction-to-decimal/2%29%20Fractions%20to%20Decimals%20Using%20Long%20Division.webp)
  ### 🛎️ What Are Terminating and Recurring Decimals?
  - **A terminating decimal** ends after a fixed number of digits.
  - **A recurring decimal** has digits that repeat forever.
  - ![Comparison of terminating and recurring decimals with examples 0.3 and 0.33333… ](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/convert-fraction-to-decimal/3%29%20Terminating%20and%20Recurring%20Decimals.webp)
  ### 🛎️ How Do We Find Recurring Decimals Using Long Division?
  - Stop dividing when you see the **same remainder again.**
  - Example: \dfrac{41}{12} = 3.4166\ldots = 3.41\dot{6}
  - ![Conversion of fractions to decimals using long division, highlighting terminating decimals (21/16 = 1.3125) and recurring decimals (41/12 = 3.416…).](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/convert-fraction-to-decimal/4%29%20Fractions%20to%20Decimals%20using%20Long%20Division.webp)
- [Adding and Subtracting Fractions and Decimals](https://maths-angel.com/lessons/adding-subtracting-fractions-and-decimals)
  > To add or subtract fractions and decimals, convert them to one form first. Compare the decimal and fraction methods with worked examples. Start free today!
  ### 🛎️ Convert Everything to One Form
  - Convert everything to **decimals** or to **fractions** before calculating.
  - Example: \dfrac{3}{5} + 0.75 = 0.6 + 0.75 = 1.35
  - ![The addition and subtraction of fractions and decimals, demonstrating both methods with worked-out solutions using 2.5 + 1/4 and 3/5 - 1.2.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/adding-subtracting-fractions-and-decimals/1%29%20Decimal%20and%20Fraction%20Method.webp)
  ### 🛎️ Which Method Should You Use?
  -	**Decimals →** faster because you can add or subtract directly
  -	**Fractions →** better for exact answers (no rounding).
  - ![Diagram comparing the decimal and fraction methods for adding and subtracting numbers with examples and conversion steps and tips.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/adding-subtracting-fractions-and-decimals/2%29%20Decimal%20vs%20Fraction%20Method.webp)
  ### 🛎️ When Should You Convert to Fractions Instead?
  - Simplify the **decimal part first** if possible (e.g. 0.55 − 0.3 = 0.25).
  - If decimals do not end neatly, **convert everything to fractions.**
  - ![Adding and subtracting fractions and decimals with conversion of decimals to fractions, finding a common denominator, and solving for the sum.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/adding-subtracting-fractions-and-decimals/3%29%20Examples%20and%20Tips.webp)
- [Multiplying Decimals](https://maths-angel.com/lessons/multiplying-decimals)
  > Multiplying decimals is done in three steps: ignore decimals, multiply as whole numbers, and place the decimal point correctly. Learn with examples. Watch free!
  ### 🛎️ How Do We Multiply Decimals?
  - First, ignore the **decimal points** and multiply as **whole numbers**.
  - Then, count the **total decimal places** and **move the decimal point left** by that number of places.
  - ![Three-step method for multiplying decimals: ignore decimal points, multiply as whole numbers, then place the decimal](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/multiplying-decimals/1%29%20Multiplying%20Decimals%20Method.webp)
  ### 🛎️ Multiply Decimals: Example
  - To find the **area**, multiply side lengths **5.5 m** and **2.6 m**.
  - Calculate **55 × 26** first, then move the **decimal point 2 places left**.
  - ![Multiplying decimals method: 0.34 × 2.5 solved by ignoring decimals (34 × 25 = 850), counting 3 decimal places, then placing the point](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/multiplying-decimals/2%29%20Multiplying%20Decimals%20Practice.webp)
  ### 🛎️ Multiplying Decimals: Application
  - In **word problems**, you often need to **multiply decimals**.
  - Multiply as **whole numbers first**, then place the **decimal point correctly**.
  - ![Real-world multiplying decimals problem: calculating cost using £6.5 per km times 8.12 km step by step](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/multiplying-decimals/3%29%20Multiplying%20Decimals%20Application.webp)
- [Calculating Fractions and Decimals](https://maths-angel.com/lessons/calculating-fractions-and-decimals)
  > The commutative, associative, and distributive laws simplify calculations with fractions and decimals. Learn to apply them with worked examples. Watch free!
  ### 🛎️ Properties of Arithmetic
  - The **commutative law** means the order does not matter: a+b=b+a and a× b=b× a.
  - The **associative law** means grouping does not matter: a+(b+c)=(a+b)+c and a×(b× c)=(a× b)× c.
  - ![Commutative, associative, and distributive law formulas for simplifying fractions and decimals](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/calculating-fractions-and-decimals/1%29%20Properties%20of%20Arithmetic%20(Formulas%29.webp)
  ### 🛎️ What Is the Commutative Law?
  - You can **swap the order** of numbers when **adding or multiplying**.
  - The **answer stays the same**.
  - ![Commutative and associative laws applied to addition with decimals and fractions, showing regrouping](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/calculating-fractions-and-decimals/2%29%20Commutative%20and%20Associative%20Law%20in%20Addition.webp)
  ### 🛎️ What Is the Associative Law?
  - You can **change the brackets** when **adding or multiplying**.
  - The **grouping changes**, but the **answer stays the same**.
  - ![Associative law in multiplication with fractions, decimals, and whole numbers worked example](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/calculating-fractions-and-decimals/3%29%20Commutative%20and%20Associative%20Law%20in%20Multiplication.webp)
  ### 🛎️ How to Simplify Multiplication with Fractions and Decimals?
  - When multiplying **decimals and fractions**, **rearrange** the numbers.
  - Group **fractions together** and **decimals together** to calculate more easily.
  - ![Distributive law applied to multiply and simplify fractions and decimals with step-by-step worked example](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/calculating-fractions-and-decimals/4%29%20Distributive%20Law%20with%20Decimals%20and%20Fractions%20.webp)
  ### 🛎️ How Does the Distributive Law Work with Multiple Terms?
  - **Factor out** the **common factor** first.
  - Simplify what is **inside the brackets**, then find the final answer.
  - ![Distributive law with multiple terms: factoring out common fractions and simplifying decimals step by step](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/calculating-fractions-and-decimals/5%29%20Distributive%20Law%20with%20More%20Terms.webp)
- [Convert Recurring Decimals to Fractions](https://maths-angel.com/lessons/recurring-decimals-to-fractions)
  > To convert recurring decimals to fractions, set x equal to the decimal, multiply, subtract, and simplify. Practise with worked examples. Start free today!
  ### 🛎️ Converting Recurring Decimals to Fractions
  - Let **x** equal the recurring decimal, for example **x = 0.333…**.
  - Multiply by **10** so the repeating digits line up, then subtract **10x − x**.
  - Solve **9x = 3**, so **x = 1/3**, meaning 0.333… = 1/3.
  - ![Step-by-step guide converting 0.333 recurring decimal to fraction 1/3 by multiplying by 10, subtracting equations, and solving for x.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/recurring-decimals-to-fractions/1%29%20Converting%20Recurring%20Decimals%20to%20Fractions%20(Method%29.webp)
  ### 🛎️ Converting Two Recurring Digits
  - Let **x = 0.4545…**, where **45** repeats.
  - Multiply by **100** so the repeating digits line up, then subtract **100x − x**.
  - Solve **99x = 45**, so **x = 45/99 = 5/11**, meaning 0.4545… = 5/11.
  - ![Converting the recurring decimal 0.4545... into a fraction, multiplying both sides by 100, subtracting the equations, and solving for x = 5/11.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/recurring-decimals-to-fractions/2%29%20Converting%20Recurring%20Decimals%20to%20Fractions%20(Example%29.webp)
  ### 🛎️ Converting Mixed Recurring Digits
  - Let **x = 0.1666…**, where only the **6** repeats.
  - Multiply by **10** and **100** so the repeating digit lines up, then subtract **100x − 10x**.
  - Solve **90x = 15**, so **x = 1/6**, meaning 0.1666… = 1/6.
  - ![Converting the recurring decimal 0.1666... into a fraction, showing multiplication by powers of 10, subtraction of equations, and solving to get 1/6.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/recurring-decimals-to-fractions/3%29%20Converting%20Recurring%20Decimals%20to%20Fractions%20(Practice%29.webp)
- [Powers and Indices](https://maths-angel.com/lessons/powers-indices)
  > Powers and indices mean repeated multiplication — for example, 5³ = 5 × 5 × 5. Covers bases, exponents, square and cube numbers, and BIDMAS. Watch free!
  ### 🛎️ What are Powers?
  - Powers show repeated multiplication of the same number.
  - For example, **3⁴** means 3 × 3 × 3 × 3.
  - ![Introduction to powers by showing the relationship that 3+3+3+3 equals 4 times 3, and 3 x 3 x 3 x 3 equals 3 to the power of 4.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/powers-indices/1%29%20Introduction%20to%20Powers%20(indices%29.webp)
  ### 🛎️ How to Read Powers
  - The **base** is the number being multiplied.
  - The **power** (small number at the top right) shows **how many times** it is multiplied.
  - ![Explanation of the base 3 and index 4 in 3^4 that it is pronounced 'three to the power of four'.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/powers-indices/2%29%20Notation%20of%20Powers.webp)
  ### 🛎️ Important Properties of Powers
  - Any number to the power of **1** equals itself (e.g. **5¹ = 5**).
  - Any number (except 0) to the **power of 0** equals **1** (e.g. **5⁰ = 1**).
  - ![Explaining key characteristics of powers, e.g., 3^4 is not the same as 4^3, a^1 is a, and for a different to 0, a^0 is 1.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/powers-indices/3%29%20Properties%20of%20Powers.webp)
  ### 🛎️ Common Powers: Squares and Cubes
  - **Square numbers** have power **2** (e.g. 4² = 16).
  - **Cube numbers** have power **3** (e.g. 3³ = 27).
  - ![Common indices chart showing square and cube numbers with examples. For example, square numbers 1, 4, 9, 16, and cube numbers 1, 8, 27, 64, 125.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/powers-indices/4%29%20Common%20Powers%20(Indices%29.webp)
  ### 🛎️ Simplifying Expressions with Powers
  - First, group the **same numbers** together in multiplication.
  - Then, write repeated multiplication using **powers.**
  - ![Cartoon blue bird explaining how to simplify 4 × 3 × 3 × 4 × 4 to 4³ × 3² using commutative and associative laws.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/powers-indices/5%29%20Simplifying%20Expressions%20with%20Powers.webp)
  ### 🛎️ Calculating with Powers
  - Following BIDMAS, work out **powers before multiplication.**
  - But if there are brackets, calculate inside the brackets first.
  - ![BIDMAS calculation rules with indices shown by comparing (2 × 5)^2 and 2 × 5^2, resulting in 100 and 50 respectively.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/powers-indices/6%29%20Calculating%20Expressions%20with%20Indices.webp)
- [Standard Form](https://maths-angel.com/lessons/standard-form)
  > Standard form writes large or small numbers as A × 10ⁿ. Learn to convert and calculate in standard form with worked examples. Watch free!
  ### 🛎️ What Is Standard Form (Scientific Notation)?
  - A way to write **very large or very small numbers** so they are easier to work with.
  - It always looks like **A × 10ⁿ**, where **A is between 1 and 10**
  - ![Standard form explained as a method of expressing large or small numbers in the form A × 10ⁿ.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/standard-form/1%29%20Standard%20Form.webp)
  ### 🛎️ How to Rewrite a Number into Standard Form?
  - Move the decimal point so the number at the front is **between 1 and 10**.
  - The number of places you move tells you the **power of 10**.
  - ![Converting 0.007 to standard form: 7 × 10⁻³ by moving the decimal three places right](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/standard-form/2%29%200.007%20to%20standard%20form.webp)
  ### 🛎️ How the Power of 10 Works?
  - Decimal moved **left → positive power** (e.g. 384000 → 3.84 × 10⁵)
  - Decimal moved **right → negative power** (e.g. 0.007 → 7 × 10⁻³)
  - ![Converting 384,000 to standard form: 3.84 × 10⁵ by moving the decimal five places left](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/standard-form/3%29%20384000%20to%20standard%20form.webp)
  ### 🛎️ How to Multiply in Standard Form?
  - Multiply the **front numbers** as normal.
  - Add the powers of 10: **10⁷ × 10⁻¹² = 10⁻⁵**
  - ![Multiplying in standard form: (2 × 10⁷) × (8 × 10⁻¹²) worked step by step](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/standard-form/4%29%20Calculating%20with%20Standard%20Form%20Example%201.webp)
  ### 🛎️ How Do We Check a Standard Form Answer?
  - The front number must be **between 1 and 10**.
  - **If not**, move the decimal and **adjust the power** (e.g. 12 × 10³ → 1.2 × 10⁴)
  - ![Calculating in standard form: 3 × 10⁻⁵ × 40,000,000 = (3 × 4) × 10⁻⁵⁺⁷ = 12 × 10² = 1.2 × 10³](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/standard-form/5%29%20Calculating%20with%20Standard%20Form%20Example%202.webp)
- [Basics of Square Roots](https://maths-angel.com/lessons/basics-of-square-roots)
  > A square root is a non-negative value that, when squared, gives the original number. Learn properties, perfect squares, and worked examples. Watch free!
  ### 🛎️ What Is a Square Root?
  - A **square root** of a number is a **non-negative number** that **multiplies by itself** to give the **original number**.
  - For example, √9 = 3 because 3 × 3 = 9.
  - ![Square roots of 9 and 25 explained as non-negative values that, when squared, give the original number](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/basics-of-square-roots/1%29%20What%20is%20a%20Square%20Roots.webp)
  ### 🛎️ Important Rules About Square Roots
  - Square roots are **non-negative**, so √x ≥ 0.
  - **Negative numbers** do **not** have square roots because no number squared is **negative**.
  - ![Square root properties: always non-negative, and negative numbers have no real square roots](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/basics-of-square-roots/2%29%20Square%20Roots%20Definition%20%26%20Properties.webp)
  ### 🛎️ Practising Square Roots
  - √0 = 0, √(144) = 12, and √(400) = 20.
  - Always check by **squaring** your answer to see if you get the **original number**.
  - ![Practising square roots with examples √0 = 0, √400 = 20, √144 = 12 and their squares shown alongside.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/basics-of-square-roots/3%29%20Practicing%20Square%20Roots.webp)
  ### 🛎️ Perfect Squares to Remember
  - **Perfect squares** are numbers made by squaring whole numbers like 1, 4, 9, 16, 25.
  - Knowing these helps you find **square roots** quickly in exams.
  - ![Practising square roots with examples including 0, 400, and 144, and a list of perfect squares from 0 to 15.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/basics-of-square-roots/4%29%20List%20of%20Common%20Square%20Roots.webp)
- [Multiplying and Dividing Square Roots](https://maths-angel.com/lessons/multiplying-dividing-square-roots)
  > Multiplying square roots follows √a × √b = √(ab), and dividing follows √a ÷ √b = √(a/b). Learn to simplify root expressions with worked examples. Watch free!
  ### 🛎️ Square Root Rules: Multiply and Divide
  - To **multiply** square roots, √a×√b=√ab.
  - To **divide** square roots, \frac{√a}{√b}=√(a/b).
  - ![Square root rules for multiplication and division showing √a × √b = √(ab) and √a ÷ √b = √(a/b)](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/multiplying-dividing-square-roots/1%29%20Multiplying%20and%20Dividing%20Roots.webp)
  ### 🛎️ Multiplying Square Roots
  - **Example 1:** √18=√(9\cdot2)=3√2.
  - **Example 2:** √(4.8)×√10=√48=4√3.
  - ![Multiplying square roots step by step: √18 simplified to 3√2 and √4.8 × √10 simplified to 4√3](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/multiplying-dividing-square-roots/2%29%20Example%20Multiplication.webp)
  ### 🛎️ Dividing Square Roots
  - **Example 1:** √(36/25)=\frac{√36}{√25}=6/5=1.2.
  - **Example 2:** \frac{√50}{√2}=√(50/2)=√25=5.
  - ![Dividing square roots examples: √(36/25) = 6/5 = 1.2 and √50 ÷ √2 = √25 = 5](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/multiplying-dividing-square-roots/3%29%20Example%20Division.webp)
- [Laws of Indices (Same Base, Same Indices)](https://maths-angel.com/lessons/laws-of-indices-same-base-or-index)
  > The laws of indices simplify powers: to multiply same base, add exponents; to divide, subtract. Covers same index rules with worked examples. Watch free!
  ### 🛎️ Indices: Base and Exponent
  - The **base** is the repeated number and the **index** (exponent) tells how many times it is multiplied.
  - Example: 2³ = 2\times2\times2.
  - ![Laws of indices showing rules for multiplying and dividing powers with same base or same index.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/laws-of-indices-same-base-or-index/1%29%20What%20are%20Indices.webp)
  ### 🛎️ Laws of Indices: Same Base Rules
  - When you **multiply** the same **base**, you **add** the **indices**, e.g. 2³\times2⁴ = 2⁷.
  - When you **divide** the same **base**, you **subtract** the **indices**, e.g. 4⁵\div4³ = 4².
  - ![Same base laws of indices: add exponents when multiplying and subtract when dividing, with examples](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/laws-of-indices-same-base-or-index/2%29%20Laws%20of%20Indices%20(Same%20Base%29.webp)
  ### 🛎️ Laws of Indices: Same Index Rules
  - When the **index** is the same, you **multiply** the **bases**, e.g. 2³\times5³ = (2\times5)³ = 10³.
  - When the **index** is the same, you **divide** the **bases**, e.g. 6⁵\div3⁵ = (6\div3)⁵ = 2⁵.
  - ![Laws of indices with same index: aⁿ × bⁿ = (ab)ⁿ for multiplying and aⁿ ÷ bⁿ = (a÷b)ⁿ for dividing, with worked examples](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/laws-of-indices-same-base-or-index/3%29%20Laws%20of%20Indices%20(Same%20Index%29.webp)
- [Negative Exponents and Power of a Power](https://maths-angel.com/lessons/negative-exponents-power-of-a-power)
  > Negative indices flip the base to a reciprocal, and power of a power multiplies the exponents. Learn the rules with clear worked examples. Watch free!
  ### 🛎️ Negative Indices: Make a Reciprocal
  - A **negative index** means take the **reciprocal**: a⁻^m=1/a^m.
  - Example: 2⁻³=1/2³=1/8.
  - ![Laws of indices showing a⁻ᵐ = 1/aᵐ and (aᵐ)ⁿ = aᵐⁿ for negative indices and power of a power](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/negative-exponents-power-of-a-power/1%29%20Law%20of%20Indices.webp)
  ### 🛎️ Negative Indices with Fractions
  - A **negative index** on a **fraction** flips it: \left(a/b\right)⁻^m=\left(b/a\right)^m.
  - Example: \left(2/3\right)⁻²=\left(3/2\right)²=9/4.
  - ![Negative indices rules shown with examples.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/negative-exponents-power-of-a-power/2%29%20Applying%20Negative%20Indices%20Rule%20on%20Fractions.webp)
  ### 🛎️ Power of a Power: Multiply the Indices
  - A **power of a power** means you **multiply** the **indices**: (a^m)ⁿ=a^mⁿ.
  - Example: (2³)⁵=2³^\^t^i^m^e^s⁵=2¹⁵.
  - ![Power of a power rule with examples using positive and negative exponents.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/negative-exponents-power-of-a-power/3%29%20Power%20of%20a%20Power%20Rules.webp)
  ### 🛎️ Power of a Power with Negative Indices
  - You still **multiply** the **indices**, so two negatives make a **positive** index.
  - Example: (3⁻²)⁻²=3⁽⁻²⁾^\^t^i^m^e^s⁽⁻²⁾=3⁴=81.
  - ![Power of a power rule illustrated with examples using positive, negative and fractional exponents.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/negative-exponents-power-of-a-power/4%29%20Power%20of%20a%20Power%20Examples.webp)
  ### 🛎️ Laws of Indices Summary
  - If the base is a **fraction** and the index is **negative**, you **flip the fraction** to make the index positive.
  - A **power of a power** means **multiply** the **indices**, even if an index is **negative**.
  - ![Laws of indices showing negative indices and power of a power, with examples including negative powers and simplification of expressions.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/negative-exponents-power-of-a-power/5%29%20Law%20of%20Indices%20and%20Power%20Rules.webp)
- [The nth Root and Fractional Indices](https://maths-angel.com/lessons/nth-root-fractional-indices)
  > Fractional indices express roots as powers. In a^(m/n), the denominator is the root, the numerator the power. Learn with clear worked examples. Watch free!
  ### 🛎️ nᵗʰ Root
  - The **nᵗʰ root** means a number that is **multiplied by itself n times**.
  - For example, the **cube root** of 27 is 3 because 3 × 3 × 3 = 27.
  - ![Square, cube, and fourth root examples showing nth root calculations and corresponding powers](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/nth-root-fractional-indices/1%29%20The%20nth%20Root.webp)
  ### 🛎️ Rules of the nᵗʰ Root
  - The **algebraic rule** is \sqrt[n]{a} × \sqrt[n]{a} × \dots = a when multiplied **n times**.
  - For example, \sqrt[3]{8} × \sqrt[3]{8} × \sqrt[3]{8} = 8.
  - ![Cube root of 8 equals 2, showing nth root rule: multiplying n roots of a number returns the original value.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/nth-root-fractional-indices/2%29%20Characteristics%20of%20the%20nth%20Root.webp)
  ### 🛎️ Roots as Fractional Indices
  - A **fractional index** is another way to write a **root**.
  - The rule is a¹^/ⁿ = \sqrt[n]{a}
  - ![Cube root of 8 equals 2, showing ∛8·∛8·∛8 = 8 and fractional indices rule a^(1/n) = ⁿ√a.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/nth-root-fractional-indices/3%29%20Basic%20Fractional%20Indices.webp)
  ### 🛎️ Example: 125¹^/³
  - The **fractional index** 1/3 means take the **cube root**.
  - 125¹^/³ = \sqrt[3]{125} = 5 because 5³ = 125.
  - ![Cube root of 125 using fractional indices: 125 to the power ⅓ = ³√125 = 5, since 5³ = 125](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/nth-root-fractional-indices/4%29%20Basic%20Fractional%20Indices%20Example.webp)
  ### 🛎️ Fractional Indices Rule
  - In a^m^/ⁿ, the **denominator** tells you the **root**.
  - The **numerator** tells you the **power**.
  - ![Fractional indices rule a^(m/n) with worked example 125^(2/3) = 25 using roots and powers](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/nth-root-fractional-indices/5%29%20Fractional%20Indices.webp)
  ### 🛎️ Example: 16³^/⁴
  - Find the **4th root** of 16 to get 2.
  - Then raise it to the **power** 3 to get 8.
  - ![Fractional indices example: 16^(3/4) step by step — fourth root of 16 is 2, then 2^3 = 8](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/nth-root-fractional-indices/6%29%20Fractional%20Indices%20Example.webp)
- [Calculating Money](https://maths-angel.com/lessons/calculating-money)
  > Calculating money involves separating pounds and pence for each operation. Learn to add, subtract, multiply, and divide money step by step. Watch free!
  ### 🛎️ Understanding Money and Prices
  - Money is measured using **pounds (£)** and **pence (p).**
  - **£1 = 100p,** for example £3.68 = 368p.
  - ![A cartoon with a sandwich priced at £3.68, illustrating UK currency conversion of pounds to pence, and showing how £3.68 equals £3 plus 68p.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/calculating-money/1%29%20Money%20and%20Prices.webp)
  ### 🛎️ How to Add Prices?
  - Add pounds and pence **separately,** then combine the total.
  - If the pence total is **100 or more,** convert to pounds (e.g. 149p = £1.49).
  - ![Calculation of adding £3.68 and £1.49 by separating pounds and pence, demonstrating the result as £5.17.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/calculating-money/2%29%20Adding%20Prices.webp)
  ### 🛎️ How to Subtract Prices?
  - Subtract pounds and pence **separately.**
  - If there is not enough pence to subtract, **exchange £1 for 100p** first.
  - ![Subtracting money in UK currency with pounds and pence, and step-by-step calculation of £10 minus £5.17 to find the difference of £4.83.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/calculating-money/3%29%20Subtracting%20Prices.webp)
  ### 🛎️ How to Multiply a Price?
  - Multiply the pounds and pence separately, then add them together.
  - **Convert extra pence into pounds** at the end if needed.
  - ![Calculating the price of 8 chocolate bars, each costing £1.20, by multiplying 8 by £1.20, resulting in £9.60.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/calculating-money/4%29%20Multiplying%20a%20Price.webp)
  ### 🛎️ How to Divide a Price?
  - Convert the full amount **into pence first** (e.g. £9.60 = 960p).
  - Divide, then **convert back** to pounds and pence (e.g. 320p = £3.20).
  - ![Calculating £9.60 divided by 3 step by step using pounds and pence, resulting in £3.20.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/calculating-money/5%29%20Dividing%20a%20Price.webp)
- [Converting and Calculating Units of Length](https://maths-angel.com/lessons/units-of-length)
  > Metric units of length include km, m, cm, and mm. Learn to convert between them and add or subtract lengths with step-by-step examples. Start free today.
  ### 🛎️ What are Metric Units of Length?
  - Length is measured in millimetres (**mm**), centimetres (**cm**), metres (**m**), and kilometres (**km**)
  - The conversion rates you have to know: **10 mm = 1 cm**, **100 cm = 1 m**, and **1000 m = 1 km**
  - ![Conversion chart for metric units of length showing millimetres, centimetres, metres, and kilometres with examples.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/units-of-length/1%29%20Converting%20Units%20of%20Length.webp)
  ### 🛎️ How to Convert Units of Length?
  - Converting to a **smaller unit** means **multiplying**; converting to a **larger unit** means **dividing**.
  - For example, **47 m = 4700 cm** (multiply by 100) and **5200 m = 5.2 km** (divide by 1000).
  - ![Converting 47 metres to 4700 centimetres and 5020 metres to 5.020 kilometres using unit conversion rules.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/units-of-length/2%29%20Converting%20Units%20of%20Length%20Example.webp)
  ### 🛎️ How to Calculate with Different Units?
  - Before adding or subtracting, **convert all values to the same unit**.
  - Choose one unit (e.g. all cm or all m) and **do not mix units**.
  - ![Adding and subtracting units of length by standardising to one unit first, showing 1.75 m + 40 cm = 2.15 m and 1.75 m - 40 cm = 1.35 m as examples.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/units-of-length/3%29%20Calculating%20Units%20of%20Length.webp)
- [Converting and Calculating Units of Mass](https://maths-angel.com/lessons/units-of-mass)
  > Metric units of mass include tonnes, kg, g, and mg. Learn to convert between them and add or subtract masses with step-by-step examples. Start free today.
  ### 🛎️ What are Metric Units of Mass?
  - Mass is measured in milligrams **(mg)**, grams **(g)**, kilograms **(kg)**, and tonnes **(t)**.
  - The conversion rates you have to know: **1000 mg = 1 g**, **1000 g = 1 kg**, and **1000 kg = 1 t**.
  - ![Units of mass conversion chart showing milligrams, grams, kilograms, and tonnes with examples: 10 mg pills, 500 g pasta, 5 kg rice, and 8 t crane.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/units-of-mass/1%29%20Converting%20Units%20of%20Mass.webp)
  ### 🛎️ How to Convert Units of Mass?
  - Converting to a **smaller unit** means **multiplying**; converting to a **larger unit** means **dividing**.
  - For example, **21 kg = 21000 g** (multiply by 1000) and **560 kg = 0.56 t** (divide by 1000).
  - ![Converting 21 kilograms to 21000 grams and 560 kilograms to 0.56 tonnes using multiplication and division by 1000.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/units-of-mass/2%29%20Converting%20Units%20of%20Length%20Example.webp)
  ### 🛎️ How to Calculate with Different Units?
  - Before adding or subtracting, **convert all values to the same unit**.
  - Choose one unit (e.g. all grams or all kilograms) and do not mix units.
  - ![Adding and subtracting units of mass by converting to one unit first, showing 3.25 kg + 30 g = 3.28 kg and 3.25 kg - 30 g = 3.22 kg as examples.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/units-of-mass/3%29%20Calculating%20Units%20of%20Mass.webp)
- [Converting Time and Calculating Time](https://maths-angel.com/lessons/converting-time)
  > 1 hour = 60 minutes and 1 minute = 60 seconds. Learn to convert between time units and calculate time differences with worked examples. Watch free!
  ### 🛎️ What are Units of Time?
  - Time is measured in **seconds** (s), **minutes** (min), **hours** (h), and **days.**
  - The conversion rates you must know: **60 s = 1 min, 60 min = 1 h,** and **24 h = 1 day.**
  - ![Conversions between units of time showing 1 day equals 24 hours, 1 hour equals 60 minutes, and 1 minute equals 60 seconds.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/converting-time/1%29%20Converting%20Units%20of%20Time.webp)
  ### 🛎️ How to Convert Units of Time?
  - Converting to a **smaller unit** means **multiplying;** converting to a **larger unit** means **dividing.**
  - For example, **2 h = 120 min** (×60) and **300 s = 5 min** (÷60).
  - ![Converting time examples with 2 hours equalling 120 minutes and 300 seconds equalling 5 minutes.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/converting-time/2%29%20Examples%20on%20Converting%20Units%20of%20Time.webp)
  ### 🛎️ How to Add Time?
  - Add minutes and seconds **separately.**
  - If seconds are 60 or more, convert **60 s into 1 min.**
  - ![Adding time: 35 min 20 s + 20 min 45 s = 55 min + 65 s, then converting 65 s to 1 min 5 s, giving 56 min 5 s](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/converting-time/3%29%20Adding%20Time.webp)
  ### 🛎️ How to Subtract Time?
  - Subtract minutes and seconds **separately.**
  - If there are not enough seconds, **exchange 1 min for 60 s** first.
  - ![STime difference: 56 min 5 s − 54 min 23 s, subtracting minutes and seconds separately with borrowing, result 1 min 42 s](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/numbers/converting-time/4%29%20Subtracting%20Time.webp)

## Algebra

- [Distributive Property](https://maths-angel.com/lessons/distributive-property)
  > The distributive property lets you multiply a number across a sum or difference. Learn to expand and factorise brackets with clear examples. Watch free!
  ### 🛎️ What is the Distributive Property?
  - The distributive property means **multiplying a number across brackets**.
  - It is written as **a × (b + c) = a × b + a × c**.
  - ![The Distributive Law for addition and subtraction, illustrated with an example showing 3×(4+2) as 3×4 + 3×2, using green and yellow balls.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/distributive-property/1%29%20The%20Distributive%20Law.webp)
  ### 🛎️ Applying the Distributive Property
  -	Multiply the number **outside the brackets** by **each term inside**.
  - This can be used to expand brackets or simplify calculations.
  - ![Applying the distributive law to expand and factorise brackets, with examples for 15 x (100 + 2) and 50 x (24 - 14) equations.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/distributive-property/2%29%20Applying%20the%20Distributive%20Law.webp)
  ### 🛎️ Distributive Property with More Than Two Terms
  - The distributive property also works with **more than two terms**.
  - For example, **a × (b + c + d) = a × b + a × c + a × d**.
  - ![The distributive law for multiplying numbers, such as 5*221, by factorising to 5*(200+20+1) and expanding to 5*200+5*20+5*1.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/distributive-property/3%29%20Distributive%20Law%20for%20Multiple%20Numbers.webp)
  ### 🛎️ Should I Use the Distributive Property with Division?
  - **Do not** use the distributive property when dividing.
  - When there is division, it is usually better to **calculate inside the brackets** first.
  - ![Explanation that the distributive law generally does not simplify division, showing the example (19+2)/3.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/distributive-property/4%29%20Remarks%20on%20the%20Distributive%20Law.webp)
- [Deriving and Evaluating Expressions with One Variable](https://maths-angel.com/lessons/expressions-with-one-variable)
  > A variable represents a changing value like x. Learn how to derive and evaluate expressions with one variable, such as 20 + 5x, with clear examples. Watch free!
  ### 🛎️ Understanding Variables
  - A **variable** is a **letter** that stands for a **number that can change**.
  - For example, **x** can represent the **number of months passed**.
  - ![Savings (£)=20+5x with x months; examples for x=1,2,3: 20+(5×1), 20+(5×2), 20+(5×3)](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/expressions-with-one-variable/1%29%20Expression%20with%20One%20Variable.webp)
  ### 🛎️ Understanding Expressions
  - An **expression** is formed by **combining numbers and variables**.
  - It shows how a value changes and has **no equals sign**.
  - ![Savings (£)=20+5x evaluated for x=8,12,20: £60=20+(5×8), £80=20+(5×12), £120=20+(5×20)](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/expressions-with-one-variable/2%29%20Evaluate%20Expression%20with%20One%20Variable.webp)
  ### 🛎️ Forming Expressions
  - Example 1: **20 + 5x** means start with 20 and **add 5 for each x**.
  - Example 2: **50 − 3y** means start with 50 and **subtract 3 for each y**.
  - ![Balance (£)=50−3y for y packs; examples y=1,2,3: £47=50−(3×1), £44=50−(3×2), £41=50−(3×3)](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/expressions-with-one-variable/3%29%20How%20to%20Form%20An%20Expression.webp)
  ### 🛎️ Evaluating Expressions
  - To **evaluate** an expression, you **replace** the variable with a **number**.
  - You then **calculate** to find the **value** of the expression.
  - ![Deriving and evaluating one-variable expression Balance(£)=50−3y, with y=10 giving £20 and y=15 giving £5 for £3 per pack.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/expressions-with-one-variable/4%29%20How%20to%20Evaluate%20An%20Expression.webp)
- [Simplifying Expressions](https://maths-angel.com/lessons/simplifying-expressions)
  > Simplifying expressions means rewriting them in a shorter form. Learn to expand brackets and combine like terms with step-by-step examples. Watch free!
  ### 🛎️ How to Simplify an Expression?
  - **Expand brackets** first using the distributive law.
  - Then **collect like terms** by adding or subtracting their coefficients.
  - ![Steps to simplify an expression by expanding brackets and collecting like terms, showing -5x + 2(3 + x) simplified to -3x + 6.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/simplifying-expressions/1%29%20How%20to%20Simplify%20Expressions.webp)
  ### 🛎️ Collecting Like Terms
  - Like terms have the **same variable** and the **same power** (e.g. 2x² and 3x²).
  - To collect like terms, add or subtract the **numbers**, then keep the **variable** the same.
  - ![Simplifying expressions by expanding brackets using distributive law, and combining like terms to simplify 3(2x - 1) + 4(-x + 5) into 2x + 17.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/simplifying-expressions/2%29%20Simplify%20Expressions%20Example.webp)
  ### 🛎️ Important Signs and Common Mistakes
  - A **minus sign before brackets** changes the sign of **every term inside**.
  - Terms with different powers (e.g. x and x²) are **not like terms** and cannot be combined.
  - ![Simplifying expressions by expanding brackets using distributive law, and combining like terms to simplify 3(2x - 1) + 4(-x + 5) into 2x + 17.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/simplifying-expressions/3%29%20Simplify%20Expressions%20Practice.webp)
- [Simplifying Expressions with Multiple Variables](https://maths-angel.com/lessons/simplifying-expressions-multiple-variables)
  > Simplify expressions with multiple variables using exponents and like terms. Learn the step-by-step process with worked examples and practice. Watch free!
  ### 🛎️ Expressions with Multiple Variables
  - An expression has **multiple variables** if it has more than one **letter**.
  - For example, a + 2b + 3c² has the variables **a**, **b**, and **c**.
  - ![Expressions with multiple variables include more than one variable, shown by the expression a + 2b + 3c squared.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/simplifying-expressions-multiple-variables/1%29%20Expression%20with%20Multiple%20Variables.webp)
  ### 🛎️ Using Exponents to Simplify
  - **Exponents** are used for **repeated multiplication**.
  - For example, x × y × y can be written as xy².
  - ![Simplification techniques for exponents showing multiplication rules with variables x and y, and explanation of the commutative law.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/simplifying-expressions-multiple-variables/2%29%20Exponents%20to%20Simplify%20Expressions.webp)
  ### 🛎️ Collecting Like Terms
  - **Like terms** have the **same variables** with the **same powers**.
  - For example, a and a² are **not like terms** and cannot be combined.
  - To collect like terms, **add or subtract the coefficients**, for example 5ab - 2ab = 3ab.
  - ![Simplification techniques showing how to collect like terms by adding coefficients and identifying like terms with the same variables and exponents.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/simplifying-expressions-multiple-variables/3%29%20Collecting%20Like%20Terms%20to%20Simplify%20Expressions.webp)
  ### 🛎️ Simplifying an Expression Step by Step
  - First **expand brackets** using the **distributive law**.
  - Then **use exponents** and **collect like terms** to simplify fully.
  - ![Simplifying algebraic expression involving multiple variables using distributive law, exponents, and collecting like terms.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/simplifying-expressions-multiple-variables/4%29%20Simplifying%20Expressions%20with%20Multiple%20Variables%20Example.webp)
- [Expanding Double Brackets](https://maths-angel.com/lessons/expanding-double-brackets)
  > Expanding double brackets uses the formula (a+b)(c+d) = ac+ad+bc+bd. Learn to multiply and simplify binomial products with worked examples. Watch free!
  ### 🛎️ The Distributive Law
  - The **distributive law** means multiplying a term by **everything inside the brackets**.
  - For example, b(c+d)=bc+bd.
  - ![Visual explanation of the distributive law showing b(c + d) equals bc + bd using coloured rectangles](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/expanding-double-brackets/1%29%20The%20Distributive%20Law.webp)
  ### 🛎️ Binomial Products: Formula
  - A **binomial product** means multiplying **two brackets** together.
  - For (a+b)(c+d), multiply **every pair** to get ac+ad+bc+bd.
  - ![Expanding double brackets showing the binomial product formula (a+b)(c+d)=ac+ad+bc+bd step-by-step. ](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/expanding-double-brackets/2%29%20Expanding%20Double%20Brackets%20(Formula%29.webp)
  ### 🛎️ Expanding Two Brackets: Example
  - Always include the **signs** when multiplying terms.
  - For example, (2x-1)(3x+2)=2x\cdot3x+2x\cdot2-1\cdot3x-1\cdot2.
  - ![Example (2x - 1)(3x + 2) for expanding double brackets, fully simplified to 6x² + x - 2.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/expanding-double-brackets/3%29%20Expanding%20Double%20Brackets%20(Example%201%29.webp)
  ### 🛎️ Expanding with Negatives
  - Rewrite subtraction as **adding a negative** before expanding.
  - For example, (-3x+2)(5-y)=(-3x)\cdot5+(-3x)·(-y)+2\cdot5+2·(-y).
  - ![Example (-3x + 2)(5 - y) for expanding double brackets, fully simplified to -15x + 3xy + 10 - 2y.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/expanding-double-brackets/4%29%20Expanding%20Double%20Brackets%20(Example%202%29.webp)
- [Square of a Binomial](https://maths-angel.com/lessons/square-of-a-binomial)
  > The square of a binomial expands (a + b)² and (a - b)². Learn the key formulas, see geometric visuals, and follow step-by-step examples. Watch free!
  ### 🛎️ Squaring a Binomial: Where the Formula Comes From
  - **Squaring** a binomial means multiplying it by **itself**, like  (a+b)(a+b) .
  - The **middle term** appears twice, which is why we get  2ab .
  - ![Expansion and simplification of (a + b)² to a² + 2ab + b² using distributive method.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/square-of-a-binomial/1%29%20Square%20of%20a%20Binomial%20Derivation.webp)
  ### 🛎️ Squaring a Binomial: See It Visually
  - The square splits into **four parts** showing  a² ,  b² , and **two**  ab  rectangles.
  - Adding the areas gives  (a+b)² = a² + b² + 2ab .
  - ![Square of a binomial formula, showing (a + b)² = a² + b² + 2ab with coloured squares and rectangles representing each term.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/square-of-a-binomial/2%29%20Square%20of%20a%20Binomial%20Visualisation.webp)
  ### 🛎️ The 3 Key Formulas
  -  (a+b)² = a² + b² + 2ab .
  -  (a-b)² = a² + b² - 2ab .
  -  (a+b)(a-b) = a² - b² .
  - ![Three binomial square formulas, including (a + b)² = a² + b² + 2ab, (a - b)² = a² + b² - 2ab, and (a + b)(a - b) = a² - b².](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/square-of-a-binomial/3%29%20Square%20of%20a%20Binomial%203%20Formulas.webp)
  ### 🛎️ Using (a+b)²: Example
  - Identify **a** and **b** including their **signs**, for example in (-2x+3)² we have **a = -2x** and **b = 3**.
  - Work out squares first, then add the end term to get (-2x)² + 3² + 2 × (-2x) × 3 = 4x² + 9 - 12x.
  - ![Binomial expansion showing the formula (a + b)² = a² + b² + 2ab, and an example (-2x + 3)² = 4x² + 9 - 12x.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/square-of-a-binomial/4%29%20Square%20of%20a%20Binomial%20Formula%201%20Example.webp)
  ### 🛎️ Using (a-b)²: Example
  - Identify **a** and **b** including the **minus sign**, for example in (5x-2y)² we have **a = 5x** and **b = 2y**.
  - Work out squares first, then subtract the end term to get (5x)² + (2y)² - 2 × 5x × 2y = 25x² + 4y² - 20xy.
  - ![Expanding the binomial (5x - 2y)² using the formula (a - b)² = a² + b² - 2ab, resulting in 25x² + 4y² - 20xy.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/square-of-a-binomial/5%29%20Square%20of%20a%20Binomial%20Formula%202%20Example.webp)
  ### 🛎️ Using (a+b)(a-b): Example
  - Check the brackets use the **same terms** with different **signs**, for example (3x+4)(3x-4).
  - Square both terms and subtract to get (3x)² - 4² = 9x² - 16.
  - ![Identifying variables a and b with signs and coefficients in the formula (a + b)(a - b) = a² - b², using (3x + 4)(3x - 4) = 9x² - 16 as an example.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/square-of-a-binomial/6%29%20Square%20of%20a%20Binomial%20Formula%203%20Example.webp)
- [Introduction to Mapping](https://maths-angel.com/lessons/introduction-to-mapping)
  > Mapping in maths shows the relationship between two quantities, like time and speed. Learn how to use tables and graphs to visualise mappings. Watch free!
  ### 🛎️ What Is a Mapping?
  - A **mapping** shows how one value **links** to another value. We say **X is mapped to Y** using an **arrow**.
  - For example, **time** is mapped to **speed**, so each **time value** has a **corresponding speed**.
  - ![Mapping time to speed with a table showing time in hours and corresponding speed in km/h, illustrating the relationship between the two quantities.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/introduction-to-mapping/1%29%20Mapping%20Definition%20and%20Example.webp)
  ### 🛎️ How to Visualise Mapping?
  - You can use a **table** and a **graph** to visualise the mapping.
  - A **table** shows exactly which **X value matches each Y value**.
  - A **graph** shows visually how **Y changes when X changes**.
  - ![Mapping of time and speed shown through a table and graph, with time in hours and speed in km/h.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/introduction-to-mapping/2%29%20Mapping%20Through%20Table%20and%20Graph.webp)
  ### 🛎️ Using Mappings in Real Life
  - **Categories**, such as **types of balls**, can be mapped to **weights**.
  - Each input and its corresponding output can be shown in a **table or graph**.
  - When the input is **discrete** categories, you **should not** join the points with a line.
  - ![Table and graph comparing the weights of different types of balls.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/introduction-to-mapping/3%29%20Mapping%20Example%20Types%20of%20Balls%20and%20Their%20Weight.webp)
- [Introduction to Formulas](https://maths-angel.com/lessons/introduction-to-formulas)
  > A formula connects variables, like y = 2x + 3. Learn how to substitute values, use tables and graphs, and apply formulas to real-life problems. Watch free!
  ### 🛎️ What Is a Formula?
  - A **formula** is a mathematical rule that links two **quantities**.
  - You **substitute x values** into a formula to find the **corresponding y values**.
  - ![Introduction to formulas showing a mathematical equation y = 2x + 3, with examples for x values of 1, 2, and 3, and the definition of a formula.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/introduction-to-formulas/1%29%20What%20is%20a%20Formula.webp)
  ### 🛎️ How to Visualise a Formula?
  - Use the formula to make a **table** of **x and y values**.
  - Then plot these values on a **graph** to see the **pattern**.
  - ![Visualising formulas using a table and a graph for the equation y = 2x + 3.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/introduction-to-formulas/2%29%20Visualising%20Formulas.webp)
  ### 🛎️ Using Formulas to Solve Problems
  - A **formula** can represent real-life situations like saving **money**.
  - You **substitute values** to solve specific **problems**.
  - ![Chart explaining how to use the formula y = 20 + 12x to calculate savings, with an example showing it takes 5 weeks to save £80.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/introduction-to-formulas/3%29%20Applying%20Formulas.webp)
- [Introduction to Functions and Graphs](https://maths-angel.com/lessons/introduction-to-functions-and-graphs)
  > A function maps each input to exactly one output, like f(x) = 2x + 1. Learn function notation, graph functions, and find values from graphs. Watch free!
  ### 🛎️ What Does Function Notation f(x) Mean?
  - f(x) shows the **output** when the **input** is x.
  - To find f(3), **substitute x = 3** into the rule.
  - For example, if f(x) = 2x + 1, then f(3) = 2(3) + 1 = 7.
  - ![Visual representation of function notation showing the function f(x) = 2x + 1 and its calculated outputs for f(1), f(2), and f(3).](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/introduction-to-functions-and-graphs/1%29%20Function%20Notation.webp)
  ### 🛎️ How to Graph a Function?
  - First **create a table** by choosing x-values and finding the matching **f(x)** values.
  - Then **plot the points** on a graph and join them with a straight or smooth line.
  - ![Steps to graph a function showing a table of values for f(x)=2x+1 and its corresponding plotted graph with points.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/introduction-to-functions-and-graphs/2%29%20Plot%20a%20Graph.webp)
  ### 🛎️ Does a Graph Represent a Function?
  - A graph is a **function** if each **x-value** has only **one y-value**.
  - If one **x-value** has **more than one y-value**, it is **not a function**.
  - ![2 graphs showing non-functions: one with a circle and another with a vertical line, explaining that one x-value must not have multiple y-values.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/introduction-to-functions-and-graphs/3%29%20Graphs%20that%20do%20not%20represent%20a%20function.webp)
  ### 🛎️ Reading Function Values from Graphs
  - To find f(x), go to the **x-value** and read the **y-value** on the graph.
  - Solving f(x) = 0 means the **y-value is zero**. This happens where the graph **crosses the x-axis**.
  - ![Graph showing function values with examples: f(-2) = -3 and f(x) = 0 at x = -4, 0, 5, with a function definition.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/introduction-to-functions-and-graphs/4%29%20Reading%20Function%20Values%20from%20Graphs.webp)
- [Solving Equations](https://maths-angel.com/lessons/solving-equations)
  > Solving equations means finding the value that makes the equation true. Learn to simplify, rearrange terms, and solve for x with clear examples. Watch free!
  ### 🛎️ Solving an Equation
  - **Solving an equation** means finding the value of **x** that makes the equation true.
  - You can **check** the solution by **substituting** the value of **x** and seeing if both sides are equal.
  - ![Solving an equation means finding the value of the unknown that makes the equation true, shown with the example x + 3 = 8, where x = 5.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/solving-equations/1%29%20What%20Does%20Solving%20Equations%20Mean.webp)
  ### 🛎️ Simplify Both Sides
  - Use the **distributive law** to expand brackets.
  - **Collect like terms** to simplify each side of the equation.
  - ![Solving a linear equation using distributive law and simplification, steps: simplify both sides, rearrange the equation, and solve for the unknown.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/solving-equations/2%29%20Solving%20Equations%20Techniques.webp)
  ### 🛎️ Rearrange and Solve
  - Move all **terms with x** to one side and all **constants** to the other side.
  - Divide by the **coefficient of x** to find the value of **x**.
  - ![Steps for solving an equation using simplification, rearrangement, and solving for the unknown variable, with an example showing the solution x = 3.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/solving-equations/3%29%20Solving%20Equations%20Example.webp)
- [Solving Rational Equations](https://maths-angel.com/lessons/solving-rational-equations)
  > Rational equations have fractions with x in the denominator. Learn to solve by clearing denominators and checking solutions with worked examples. Watch free!
  ### 🛎️ Rational Equations
  - Rational equations have **x in the denominator**, so extra care is needed.
  - Multiply **every term on both sides** by the **denominator containing x**.
  - This turns the equation into a **normal equation without fractions**.
  - ![Steps to solve an equation with x in the denominator, including isolating x, clearing the denominator, and checking the solution.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/solving-rational-equations/1%29%20How%20to%20Solve%20Rational%20Equations%20Step%20by%20Step.webp)
  ### 🛎️ Solve and Check
  - **Rearrange the equation** to find the value of **x**.
  - **Check the solution** by substituting it back into the **original equation**.
  - Make sure the value of **x** does **not make the denominator zero**.
  - ![Solving equations with rational expressions where x is in the denominator, step-by-step process with a worked example and solution x = 1.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/solving-rational-equations/2%29%20Solving%20Equations%20with%20X%20in%20Denominator%20(Example%29.webp)
- [Gradient and Y-Intercept in Linear Equations](https://maths-angel.com/lessons/m-and-c-in-linear-equations)
  > In y = mx + c, m is the gradient and c is the y-intercept. Learn what m and c mean and how to draw linear equations on a graph with examples. Watch free!
  ### 🛎️ Recap: Linear Equations
  - Linear equations are usually written in the form **y = mx + c**.
  - When you draw a linear equation, it makes a **straight line** on a graph.
  - ![Linear equation y = mx + c showing the gradient (m) and y-intercept (c), with a graph of a straight line.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/m-and-c-in-linear-equations/1%29%20m%20and%20c%20in%20Linear%20Equations.webp)
  ### 🛎️ The y-intercept (c)
  - **c** is the **y-intercept**, where the line crosses the **y-axis**.
  - This is the value of **y** when **x = 0**.
  - ![Linear equation y=mx+c with highlighted y-intercept c where line crosses y-axis. Diagram shows lines with different y-intercepts, one passing origin.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/m-and-c-in-linear-equations/2%29%20Intercept%20Linear%20Equations.webp)
  ### 🛎️ The Gradient (m)
  - **m** is the **gradient**, which shows how much **y changes** when **x increases by 1**.
  - If **m > 0** the line goes **upwards**, and if **m < 0** the line goes **downwards**.
  - A larger **|m|** means the line is **steeper**.
  - ![Graph illustrating y = mx + c with explanations of gradient (m) and y-intercept (c), and examples of positive and negative gradients.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/m-and-c-in-linear-equations/3%29%20Slope%20of%20Linear%20Equations.webp)
  ### 🛎️ Drawing a Linear Equation: Step 1
  - Start by **plotting the y-intercept** using the value of **c**.
  - This gives you the **first point** on the line.
  - ![How to draw a linear equation with gradient (m) and y-intercept (c) shown on a graph, step-by-step guide with illustrations.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/m-and-c-in-linear-equations/4%29%20Draw%20a%20Linear%20Equation%20y%20%3D%202x%20%2B%201.webp)
  ### 🛎️ Drawing a Linear Equation: Step 2
  - Use the **gradient m** to find a **second point**.
  - For example, if **m = -3**, go **1 right and 3 down** from the first point.
  - **Join the two points** to draw the straight line.
  - ![Steps to draw a linear equation y = -3x + 4, find y-intercept at 4, move 1 step right and 3 steps down, and connect points with the line.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/m-and-c-in-linear-equations/5%29%20Draw%20a%20Linear%20Equation%20y%3D-3x%2B4.webp)
- [Finding the Equation of a Straight Line](https://maths-angel.com/lessons/equation-of-a-straight-line)
  > The equation of a straight line is y = mx + c, where m is the gradient and c is the y-intercept. Learn to find both using two points. Watch free!
  ### 🛎️ The Straight Line Formula: y = mx + c
  - **m** is the **gradient** and tells you how steep the line is.
  - **c** is the **y-intercept** and shows where the line crosses the **y-axis**.
  - ![Linear equation y = mx + c with m labelled as slope and c as y-intercept, shown on a graph crossing the y-axis.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/equation-of-a-straight-line/1%29%20Linear%20Equations.webp)
  ### 🛎️ Finding a Linear Equation Using Two Points
  - The **gradient** is found using **m = (y₂ − y₁) ÷ (x₂ − x₁)**.
  - For example, with points **(2,3)** and **(4,7)**, the **gradient** is **m = (7 − 3) ÷ (4 − 2) = 2**.
  - ![Finding linear equation y = mx + c using two points, and steps to calculate gradient (m) and y-intercept (c) with points (2,3) and (4,7) on a graph.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/equation-of-a-straight-line/2%29%20Determining%20Linear%20Equations.webp)
  ### 🛎️ Checking If a Point Lies on a Line
  - Substitute **(3,5)** into **y = 2x − 1** to check if **5 = 2 × 3 − 1**.
  - Because **5 = 5**, the point **lies on the line**.
  - ![Verifying if points (3, 5) and (-2, -4) lie on the line y = 2x - 1 by substituting coordinates into the equation illustrating right and wrong results.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/equation-of-a-straight-line/3%29%20Checking%20whether%20point%20on%20Line.webp)
- [How to Find X-Intercept, Y-Intercept and Intersections](https://maths-angel.com/lessons/x-intercept-y-intercept-intersection)
  > Learn how to find the x-intercept (set y = 0), y-intercept (set x = 0), and where two linear equations intersect, with clear worked examples. Watch free!
  ### 🛎️ Finding Intercepts from an Equation
  - For **y = 200 − 50x**, setting **x = 0** gives a **y-intercept** of **200**.
  - Setting **y = 0** and solving **0 = 200 − 50x** gives the **x-intercept** of **4**.
  - ![Diagram explaining the x- and y-intercepts of linear equations using the equation y = 200 - 50x, illustrating where the line crosses the axes.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/x-intercept-y-intercept-intersection/1%29%20Intercepts%20of%20Linear%20Equations.webp)
  ### 🛎️ Intersections of Linear Equations
  - An **intersection** is the point where **two lines meet**.
  - At the intersection, both equations have the **same x and y values**.
  - ![Intersection of two linear equations, y = 200 - 50x and y = 150 - 30x, at point (2.5, 75) with x-axis as hours drove and y-axis as remaining distance.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/x-intercept-y-intercept-intersection/2%29%20Intersection%20of%20Linear%20Equations.webp)
  ### 🛎️ Finding the Intersection Point
  - Set the two equations **equal to each other** to find **x**.
  - Put the value of **x** back into any equation to find **y**.
  - ![Steps to find the intersection of two linear equations by setting equations equal and solving for x and y, with equations and example values.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/x-intercept-y-intercept-intersection/3%29%20Determining%20the%20Intersection.webp)
- [Simple Quadratic Equations](https://maths-angel.com/lessons/simple-quadratic-equations)
  > The coefficient a in y = ax² controls whether a parabola opens up or down and how wide it is. Learn with visual examples and practice questions. Watch free!
  ### 🛎️ What Is a Quadratic Equation?
  - A **quadratic equation** has an **x² term**.
  - Its graph is called a **parabola**.
  - ![Quadratic equation formula y = ax² with a not equal to zero, highlighting coefficient 'a'.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/simple-quadratic-equations/1%29%20Simple%20Quadratic%20Equation.webp)
  ### 🛎️ Graphing the Standard Parabola y = x²
  - You find points by **squaring x** to get **y**.
  - The graph is **symmetrical** and has a **minimum** at the **origin**.
  - Points like **(-2,4)** and **(2,4)** show the symmetry.
  - ![Graph of the quadratic equation y = x² with a table of x and y values and a standard parabola on a coordinate plane where a = 1.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/simple-quadratic-equations/2%29%20Standard%20Parabola.webp)
  ### 🛎️ Effect of a in y = ax²
  - The **sign of a** decides if the parabola opens **upwards** or **downwards**.
  - The **size of |a|** controls the **width** of the parabola.
  - A **larger |a|** makes the parabola **narrower**.
  - ![Graph illustrating how the sign of 'a' affects direction and its absolute value affects the width of the parabola.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/simple-quadratic-equations/3%29%20Effect%20of%20%27a%27%20in%20quadratic%20equation.webp)
- [Vertex Form and Parabola Transformations](https://maths-angel.com/lessons/vertex-form)
  > Vertex form is y = a(x - h)² + k, where (h, k) is the vertex of the parabola. Learn to find the vertex and transform parabolas with examples. Watch free!
  ### 🛎️ Vertex Form of a Quadratic
  - The **vertex form** is **y = a(x − h)² + k**, and the **vertex** is **(h, k)**.
  - For example, in **y = 2(x + 3)² − 1**, the vertex is **(−3, −1)**.
  - ![Vertex form of quadratic equation y = a(x - h)² + k, showing the vertex at point (h,k) on a graph of a parabola.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/vertex-form/1%29%20Vertex%20Form%20of%20Quadratic%20Equations.webp)
  ### 🛎️ What Do h and k Mean?
  - **h** moves the graph **left or right**.
  - **k** moves the graph **up or down**.
  - ![Vertex form of quadratic equation y = a(x - h)² + k showing how horizontal (h) and vertical (k) shifts affect a parabola's position on a graph.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/vertex-form/2%29%20How%20to%20Find%20the%20Vertex%20of%20a%20Quadratic%20Function.webp)
  ### 🛎️ Shifting a Parabola Using Vertex Form
  - Start with **y = 2x²**, which has its **vertex at (0,0)**.
  - Moving the **vertex** **3 left** and **1 down** gives the new graph **y = 2(x + 3)² − 1**.
  - ![Graph showing transformation of the quadratic equation from y = -2x² with vertex (0,0) to y = -2(x+2)² + 3 with vertex (-2,3), shifting the parabola.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/vertex-form/3%29%20Transformations%20of%20Quadratic%20Functions%20Using%20the%20Vertex.webp)
- [Converting Quadratics: Standard Form and Vertex Form](https://maths-angel.com/lessons/converting-standard-form-and-vertex-form)
  > Convert quadratic equations between standard form and vertex form. Master completing the square and expanding brackets with step-by-step examples. Watch free!
  ### 🛎️ What are Standard Form and Vertex Form?
  - **Standard (general) form:** y = ax² + bx + c
  - **Vertex form:** y = a(x - h)² + k
  - ![Converting quadratic equations between general form y = ax² + bx + c and vertex form y = a(x - h)² +  k, where a ≠ 0.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/converting-standard-form-and-vertex-form/1%29%20General%20Form%20and%20Vertex%20Form.webp)
  ### 🛎️ Converting Vertex Form to Standard Form
  - **Expand the square** using (a + b)² = a² + 2ab + b², then expand the brackets
  - Don't forget the **constant** at the end (this is a very common mistake)
  - ![Converting a quadratic equation from vertex form, y = 2(x - 3)² - 5, to general form by expanding and simplifying to y = 2x² - 12x + 13.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/converting-standard-form-and-vertex-form/2%29%20Converting%20Vertex%20to%20General%20Form.webp)
  ### 🛎️ Converting Standard Form to Vertex Form
  - **Only factor out** from the x² and x terms. Ignore the constant
  - E.g. 3x² + 18x → 3(x² + 6x)
  - ![Conversion of a quadratic equation from general form y = 3x² + 18x + 25 to vertex form by completing the square, resulting in y = 3(x + 3)² - 2.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/converting-standard-form-and-vertex-form/3%29%20Converting%20General%20to%20Vertex%20Form.webp)
  ### 🛎️ Completing the Square
  - When completing the square, anything added inside **must be undone outside**
  - E.g. x² - 4x = (x - 2)² - 2²
  - ![Converting the quadratic equation y = -2x² + 8x - 5 from general to vertex form using completing the square method, resulting in y = -2(x - 2)² + 3.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/converting-standard-form-and-vertex-form/4%29%20Converting%20General%20to%20Vertex%20Form.webp)
- [How to Write Quadratic Equations in Vertex Form and Standard Form](https://maths-angel.com/lessons/quadratic-equations-in-vertex-form-standard-form)
  > Write quadratic equations in vertex form using the vertex and a point, or in standard form using three points. Learn both methods with clear examples. Watch free!
  ### 🛎️ Forms of **Quadratic Equations**
  - Quadratics can be written in **vertex form** or **general form**.
  - Use **vertex form** when you know the **vertex**, and **general form** when you know **points on the parabola**.
  - ![Quadratic equations with vertex form y = a(x - h)² + k and general form y = ax² + bx + c, labelled as methods for forming quadratic equations.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/quadratic-equations-in-vertex-form-standard-form/1%29%20Determining%20Quadratic%20Equations.webp)
  ### 🛎️ Using the **Vertex Form**
  - If the **vertex** is (2, 3), start with y = a(x − 2)² + 3.
  - Substitute the **point** (3, 1) to make an equation and find the value of **a**.
  - ![Forming a quadratic equation using vertex form with equation y = -2(x - 2)² + 3, where the vertex is (2,3) and the curve passes through (3,1).](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/quadratic-equations-in-vertex-form-standard-form/2%29%20Determining%20quadratic%20equations%20through%20vertex%20form.webp)
  ### 🛎️ Using the **General Form**
  - **Substitute three known points** to make **three equations**.
  - **Solve the equations together** using **substitution or elimination** to find **a**, **b**, and **c**.
  - ![Forming quadratic equations using the general form y = 2x² - x + 3, solving coefficients with three given points: (0,3), (1,4), and (2,9).](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/quadratic-equations-in-vertex-form-standard-form/3%29%20Determining%20quadratic%20equations%20through%20General%20Form.webp)
  ### 🛎️ Applying Quadratics to a **Real-Life Situation**
  - The equation y = −5/9 x² + 5 models the **shape of a bridge**.
  - Substituting x = 1 shows the **height** is above 4 m, so the truck can drive through.
  - ![Solving the classic tunnel problem. The parabola equation is y = -5/9 · x² + 5 and the vertex is at (0,5). When x = 1, the truck can pass through.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/quadratic-equations-in-vertex-form-standard-form/4%29%20Application%20of%20Quadratic%20Equations.webp)
- [Solving Simple Quadratic Equations](https://maths-angel.com/lessons/solving-simple-quadratic-equations)
  > Simple quadratic equations like x² = k and x² − dx = 0 are solved by taking square roots or factorising. Learn to spot when there is no real solution. Watch free!
  ### 🛎️ Solving Equations of the Form x² = k
  - If x² = k, take the **square root** to get x = ±√k. Important: this only works when k ≥ 0.
  - For example, solving x² = 9 gives x = ±√9, so the solutions are x = 3 and x = -3.
  - ![Solving the quadratic equation x² = k, showing the general solution x equals plus or minus the square root of k, and example of x² = 9 with solutions.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/solving-simple-quadratic-equations/1%29%20Solving%20Simple%20Quadratics.webp)
  ### 🛎️ Example: Solving 3x² - 12 = 0
  - Rearrange by **adding 12** to both sides so it becomes 3x² = 12, which is now in the form x² = k.
  - **Divide both sides by 3** to simplify and get x² = 4.
  - Take the **square root** to get x = ± 2, so the solutions are x = 2 and x = -2.
  - ![Solving quadratic equations with the steps to isolate x² and find solutions x sub one equals the positive square root of k.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/solving-simple-quadratic-equations/2%29%20Solving%20Quadratic%20Equations%20Example.webp)
  ### 🛎️ When There Is No Real Solution
  - If x² equals a **negative number**, there is **no real solution** because no real number squares to a negative.
  - For example, x² = -25 has **no real solution**, so -5 is **not** an answer.
  - ![Solving quadratic equations, demonstrating x² = k and x² + 50 = 0 leading to no real solution since x² = -25.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/solving-simple-quadratic-equations/3%29%20Solving%20Quadratic%20Equations%20No%20Solution.webp)
  ### 🛎️ Solving Equations of the Form x² - dx = 0
  - Factorise to get x(x - d) = 0.
  - Set each factor equal to **zero** to get x = 0 and x = d.
  - ![Solving quadratic equations by factorisation, showing x² - dx = 0 with solutions x sub one equals zero, and x sub two equals d.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/solving-simple-quadratic-equations/4%29%20Solving%20Quadratic%20Equations%20Factorising.webp)
  ### 🛎️ Example: 5x² = 25x
  - First make the **coefficient** of x² equal to 1 by **dividing by 5**, which helps to simplify the calculations.
  - Factorise to get x(x - 5) = 0, so the solutions are x = 0 or x = 5.
  - ![Solving quadratic equation x² - dx = 0 with steps to simplify and factorise, showing solutions x₁ = 0 and x₂ = 5.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/solving-simple-quadratic-equations/5%29%20Solving%20Quadratic%20Equations%20Example.webp)
  ### 🛎️ Solving Simple Quadratic Equations (Summary)
  - If an equation is in the form **x² = k**, the solutions are **x = ±√k** and this only works when **k ≥ 0**.
  - If an equation is in the form **x² - dx = 0**, factorise to **x(x-d)=0** and the solutions are **x=0** and **x=d**.
  - ![Solving simple quadratic equations: x² = k gives x = ±√k, and x² − dx = 0 factors to x = 0 or x = d.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/solving-simple-quadratic-equations/6%29%20Overview%20Solving%20Simple%20Quadratic%20Equations.webp)
- [Solving Quadratic Equations by Factorising](https://maths-angel.com/lessons/solving-quadratic-equations-factorising)
  > Factorising solves x² + bx + c = 0 by splitting it into (x + m)(x + n) = 0, then setting each bracket to zero. Learn with step-by-step examples. Watch free!
  ### 🛎️ Preparing the Equation
  - Make sure the equation is written as x² + bx + c = 0
  - Rearrange first if needed so **the equation is equal to zero**
  - ![Solving quadratic equations by factorisation step-by-step, note finding two numbers that multiply to the constant and add up to the coefficient of x.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/solving-quadratic-equations-factorising/1%29%20Factorisation%20Method.webp)
  ### 🛎️ Factorising the Quadratic
  - Find two numbers m and n that **multiply to c** and **add to b**
  - Write the equation as (x + m)(x + n) = 0
  - Example: (x + 3)(x + 4) = 0
  - ![Solving quadratic equations by factorisation, with example x² + 7x + 12 = 0, note steps to find factors 3 and 4, with solutions.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/solving-quadratic-equations-factorising/2%29%20Solving%20Quadratic%20Equations.webp)
  ### 🛎️ Finding the Solutions
  - Set **each bracket equal to zero**: (x - 3) = 0 or (x + 6) = 0
  - Solve to **find both values of x**: x = 3 and x = -6
  - ![Solving quadratic equations by factorisation: find two numbers that multiply to be the constant (-18) and add up to be the coefficient of x (3).](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/solving-quadratic-equations-factorising/3%29%20Practice%20Solving%20Quadratic%20Equations%20by%20Factorising.webp)
- [Solving Quadratic Equations: Quadratic Formula](https://maths-angel.com/lessons/quadratic-formula)
  > The quadratic formula solves equations in the form ax² + bx + c = 0. Learn how the discriminant b² − 4ac shows the number of solutions with examples. Watch free!
  ### 🛎️ The Quadratic Formula and the Discriminant
  - For any quadratic in the form ax²+bx+c=0, the **solutions** are given by x=\frac{-b±√(b²-4ac)}{2a}.
  - The part **under the square root**, b²-4ac, is called the **discriminant** and tells you how many solutions there are.
  - If the **discriminant** is **negative**, there are **no real solutions**, so you can stop calculating.
  - ![Quadratic formula for solving quadratic equations, and the discriminant  for determining the number of real roots.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/quadratic-formula/1%29%20The%20Quadratic%20Formula.webp)
  ### 🛎️ Example: Using the Quadratic Formula
  - For 3x²-5x+2=0, substitute a=3, b=-5, and c=2 into the formula.
  - The **discriminant** is (-5)² - 4 × 3 × 2 = 25 - 24 = 1, which means there are **two different real solutions**.
  - Substituting gives x=5\pm1/6, so the solutions are x=1 and x=2/3.
  - ![Quadratic formula example with a = 3, b = 5, c = 2 solving 3x² − 5x + 2 = 0, giving roots x₁ = 1 and x₂ = 2/3.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/quadratic-formula/2%29%20Example%20of%20How%20to%20Use%20the%20Quadratic%20Formula.webp)
  ### 🛎️ Example: Rearranging Before Using the Formula
  - First rearrange 6=-4x²+14x into the form ax²+bx+c=0: 4x²-14x+6=0.
  - Simplify by **dividing every term by 2** to get 2x²-7x+3=0, which makes the calculations much easier.
  - Substitute a=2, b=-7, and c=3 into the quadratic formula to find the solutions.
  - ![Solving a quadratic equation using the quadratic formula, with step-by-step breakdown and final solutions.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/quadratic-formula/3%29%20How%20to%20Apply%20the%20Quadratic%20Formula.webp)
- [Solving Equations with 2 Variables](https://maths-angel.com/lessons/equations-with-2-variables)
  > Equations with 2 variables represent relationships like x + 2y = 20. Learn to solve them using substitution and graphing with worked examples. Watch free!
  ### 🛎️ Equations with Two Variables
  - An equation like **x + 2y = 20** has **two unknowns**.
  - There are **many solutions** because lots of number pairs can work.
  - ![Using the equation x + 2y = 20 to illustrate prices of items with various solutions for variables x and y, illustrating multiple solutions.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/equations-with-2-variables/1%29%20Equations%20with%202%20Variables.webp)
  ### 🛎️ Solving by Substitution
  - **Substitute** the value you know into the equation to **solve** for the other variable.
  - For example, if **x = 10**, substitute it in **x + 2y = 20** to get **10 + 2y = 20**. Solving for **y**, we get **y = 5**.
  - ![Solving the equation 10 + 2y = 20 by plugging in x = 10 and finding y = 5, illustrating linear equations in two variables.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/equations-with-2-variables/2%29%20Solving%20Equation%20with%202%20variables.webp)
  ### 🛎️ Solving Graphically
  - Rewrite the equation **x + 2y = 20** as **y = −0.5x + 10** to draw the line.
  - This means **every point on the line** is a solution to the equation.
  - The point **(8, 6)** is a solution because it lies **on the line**.
  - ![Graphical solution of the system of equations y = -0.5x + 10 and x + 2y = 20 with highlighted intersection point (8, 6).](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/equations-with-2-variables/3%29%20Solving%20Equations%20with%202%20Variables%20Graphically.webp)
- [Solving Simultaneous Equations Graphically](https://maths-angel.com/lessons/solving-simultaneous-equations-graphically)
  > Simultaneous equations can be solved by plotting lines and finding where they intersect. Learn to identify one, none, or infinite solutions. Watch free!
  ### 🛎️ Systems of Linear Equations
  - When **x + y = 20** and **x − y = 10** form a **system**, they are solved **together**.
  - We look for **x and y values** that make **both equations true** at the same time.
  - ![System of linear equations representing Sebastian's and Sarah's ages with equations x + y = 20 and x - y = 10.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/solving-simultaneous-equations-graphically/1%29%20Simultaneous%20Equations.webp)
  ### 🛎️ Solving a System Graphically
  - **Rearrange** both equations into **y = mx + c** form.
  - **Plot both lines** and find where they **intersect**.
  - The **point of intersection** is the **solution of the system**.
  - ![Solving a system of equations graphically, transforming and plotting the equations y = -x + 20 and y = x - 10 to find their intersection at (15, 5).](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/solving-simultaneous-equations-graphically/2%29%20Finding%20the%20Solution%20of%20System%20of%20Equations.webp)
  ### 🛎️ Solutions of a Linear System
  - There is **one solution** if the lines **intersect once**.
  - There is **no solution** if the lines are **parallel**.
  - There are **infinite solutions** if the lines **overlap exactly**.
  - ![Three graphical linear equations showing one solution (intersection), no solution (parallel lines), and infinite solutions (overlapping lines).](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/solving-simultaneous-equations-graphically/3%29%20Possible%20Solutions%20in%20Simultaneous%20Equations.webp)
- [Simultaneous Equations: Equal Values and Substitution Method](https://maths-angel.com/lessons/equal-values-and-substitution-method)
  > Solve simultaneous equations using the equal values and substitution methods. Learn to isolate variables and find solutions with clear examples. Watch free!
  ### 🛎️ Equal Values Method
  - **Make the same variable** (like **y**) the subject in **both equations**.
  - **Set the two expressions equal** and **solve** for **x**.
  - **Substitute** the **x value** back to find **y**.
  - ![Equal Values Method steps to solve equations by isolating the same variable, with example 2x + y = 8 and x - y = 1, resulting in x = 3 and y = 2.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/equal-values-and-substitution-method/1%29%20Equal%20Value%20Methods.webp)
  ### 🛎️ Substitution Method
  - **Rearrange** one equation to **express one variable in terms of the other**, for example **y = x − 1**.
  - **Substitute** this into the other equation so it has **only one variable**, then **solve**.
  - **Exam tip:** always **substitute both values** back into the **original equations** to check.
  - ![Steps to solve a system of equations using the substitution method, expressing one variable in terms of another and verifying the solution.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/equal-values-and-substitution-method/2%29%20Substitution%20Method.webp)
  ### 🛎️ Choosing the Right Method
  - Use **equal values** when the **same variable** is easy to make the subject.
  - Use **substitution** when one equation is easy to rearrange to **remove one variable**.
  - Both methods give the **same solution** for **x and y**.
  - ![Methods for solving linear systems using Equal Values and Substitution methods, showing steps to isolate variables and solve equations.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/equal-values-and-substitution-method/3%29%20Equal%20Values%20vs%20Substitution%20Method.webp)
- [Elimination Method for Solving Simultaneous Equations](https://maths-angel.com/lessons/elimination-method)
  > The elimination method removes one variable from simultaneous equations by adding or subtracting. Learn the steps to find solutions with examples. Watch free!
  ### 🛎️ Elimination Method
  - The **elimination method** removes **one variable** by **adding or subtracting** two equations.
  - The aim is to get an equation like **4x = 12** that is easy to **solve**.
  - ![The definition of the elimination method is that it removes one variable (x or y) by adding or subtracting two equations.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/elimination-method/1%29%20The%20Elimination%20Method.webp)
  ### 🛎️ Elimination by Addition
  - **Add** the equations when one variable has **opposite signs**, like **+y** and **−y**.
  - That variable **cancels out**, leaving an equation with **one variable**.
  - ![Demonstrating the elimination method in solving simultaneous equations by adding or subtracting variables x and y.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/elimination-method/2%29%20Elimination%20through%20Addition.webp)
  ### 🛎️ Preparing the Equations
  - **Transform** the equations so one variable can be **cancelled**, like **+3y** and **−3y**.
  - **Exam tip:** when you multiply, **multiply every term on both sides** of the equation.
  - ![Solving simultaneous equations using the elimination method with subtraction steps.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/elimination-method/3%29%20Elimination%20through%20Transform%20and%20Addition.webp)
  ### 🛎️ Elimination by Subtraction
  - **Subtract** the equations when one variable has the **same sign**, like **+2x** and **+2x**.
  - This **removes** that variable so you can **solve**.
  - ![Steps for solving simultaneous equations using the elimination method with subtraction, illustrated with equations.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/elimination-method/4%29%20Elimination%20Through%20Subtraction.webp)
  ### 🛎️ Finishing and Checking
  - **Substitute** the value back to find the **other variable**.
  - **Exam tip:** always **check** by substituting both **x and y** into the **original equations**.
  - ![Elimination method for solving simultaneous equations, and steps of transform, eliminate, and solve with addition and subtraction examples.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/algebra/elimination-method/5%29%20Addition%20vs%20Subtraction%20Method.webp)

## Ratio, Proportion & Rates of Change

- [Introduction to Ratios](https://maths-angel.com/lessons/ratios)
  > A ratio compares quantities to show their relative sizes, written as a:b. Learn to simplify, expand, and divide amounts using ratios. Watch free!
  ### 🛎️ What Is a Ratio?
  - A ratio **compares quantities** and is written as **a : b.**
  - The **order matters** (apples : bananas = **2 : 3,** but bananas : apples = **3 : 2**).
  - ![Illustration for ratios with 2 apples and 3 bananas as an example. The ratio of apples to bananas is 2:3, and the ratio of bananas to apples is 3:2.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/ratios/1%29%20Ratios%20Definition%20and%20Examples.webp)
  ### 🛎️ Working with Ratio 5:2
  - A **ratio** of **5 : 2** means for every 5 parts of flour, there are 2 parts of sugar.
  - To find one part, divide **300 g** by **5** to get **60 g** per part.
  - Multiply **60 g** by **2** to get **120 g** of sugar needed.
  - ![Ratio calculation showing flour to sugar ratio of 5:2 with a worked example of 300 g of flour, determining that 120 g of sugar is needed.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/ratios/2%29%20Calculating%20with%20Ratios%20(Finding%20Parts%29.webp)
  ### 🛎️ Sharing a Whole in a Ratio
  - Add the numbers in the ratio to find the **total parts.** For example, 7:3 means 7 + 3 = 10 parts.
  - Each part = **total amount ÷ total parts.** For example, £200 total divided by 10 parts = £20 per part.
  - One person gets 7 parts, so **£20 × 7 = £140**. The other person gets 3 parts, so **£20 × 3 = £60**.
  - ![An example of sharing £200 in the ratio 7:3, total parts calculated as 10, each part worth £20. One person receives £140, the other receives £60.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/ratios/3%29%20Calculating%20with%20Ratios%20(Dividing%20a%20Whole%29.webp)
  ### 🛎️ Simplifying and Expanding Ratios
  - To **simplify,** divide every part by the **same number.** For example, **24:40** can be simplified by dividing both by 8 to get **3:5**.
  - To **expand,** multiply every part by the **same number.** For example, **3:7** can be expanded by multiplying both by 5 to get **15:35**.
  - ![Simplifying the ratio 24:40 by dividing both terms to get 3:5 and expanding the ratio 3:7 by multiplying both terms to get 15:35.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/ratios/4%29%20Simplifying%20and%20Expanding%20Ratios.webp)
- [How to Combine Ratios](https://maths-angel.com/lessons/how-to-combine-ratios)
  > Combine two ratios into one three-part ratio by matching the common term. Learn when to use LCM with step-by-step worked examples. Watch free!
  ### 🛎️ Same Middle Number
  - If the middle number is already the same, **combine the ratios directly.**
  - Example: if a:b = 2:5 and b:c = 5:3, then a:b:c = 2:5:3.
  - ![Ratio combination showing a to b = 2 to 5 and b to c = 5 to 3, with final combined ratio a to b to c = 2 to 5 to 3.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/how-to-combine-ratios/1%29%20How%20to%20Combine%20Ratios%20With%20Same%20Common%20Factor.webp)
  ### 🛎️ Different Middle Numbers
  - If the middle numbers are different, **make them the same first** using the LCM.
  - Example: a:b = 3:5 and b:c = 2:7 → make both have b = 10.
  - ![How to combine ratios when the middle term is different, a : b = 3 : 5 and b : c = 2 : 7, with final combined ratio a : b : c = 6 : 10 : 35](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/how-to-combine-ratios/2%29%20How%20to%20Combine%20Ratios%20Using%20LCM%20of%20Common%20Factor.webp)
  ### 🛎️ Writing the Combined Ratio
  - Once the middle numbers match, write **all three parts together** as one ratio.
  - Example: 8:6 and 6:15 combine to give 8:6:15.
  - ![Combined ratios calculation of water to cement (4:3) and cement to sand (2:5) to get final ratio 8:6:15.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/how-to-combine-ratios/3%29%20Combining%20Ratios%20Applications%20Question%201.webp)
  ### 🛎️ What the Combined Ratio Means
  - A combined ratio shows **how three quantities compare** at the same time.
  - Example: 24:18:45 means for every **24** of the first amount, there are **18** of the second and **45** of the third.
  - ![Example of combined ratios to find sand needed when water to cement is 4:3 and cement to sand is 2:5, with 24 buckets of water.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/how-to-combine-ratios/4%29%20Combining%20Ratios%20Applications%20Question%202.webp)
- [Fractions and Percentages](https://maths-angel.com/lessons/fractions-and-percentages)
  > Percentages represent fractions out of 100. Learn how to convert between fractions and percentages and find percentages of a total. Watch free!
  ### 🛎️ What Is a Percentage?
  - A percentage means **“out of 100”** (e.g. 70% = 70/100).
  - It is another way of writing a **fraction with denominator 100.**
  - ![Introduction to percentages with steps for converting percentages to fractions and real-life examples of percentage use.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/fractions-and-percentages/1%29%20Percentages.webp)
  ### 🛎️ Converting Fractions to Percentages
  - Change the fraction so the **denominator is 100.** For example, 6/25 = 24/100 = 24%.
  - You can do this by **expanding or simplifying** the fraction.
  - ![Converting fractions to percentages using expanding and simplifying fractions methods.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/fractions-and-percentages/2%29%20Converting%20Fractions%20to%20Percentages.webp)
  ### 🛎️ Finding a Percentage of a Total
  - To find a percentage, **divide the part by the total** and multiply by 100 (e.g. 8/20 = 40%).
  - All parts together **should add up to 100%** (e.g. 40% + 60% = 100%).
  - ![Calculating percentages using footballs and basketballs as examples, showing 40% of 20 balls are footballs and 60% are basketballs.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/fractions-and-percentages/3%29%20Percentages%20Example.webp)
- [How to Find Percentage of a Number](https://maths-angel.com/lessons/percentage-of-a-number)
  > Find a percentage of a number by multiplying the value by the rate (e.g. 30% of 200 = 60). Learn the formula plus quick tricks for 10% and 1%. Watch free!
  ### 🛎️ The Percentage Formula
  - Percentage of a **number = original number × percentage**
  - Example: 30% of 200 = 200 × 0.3 = 60
  - ![Example of finding 30% of 200 using the formula base value × percentage rate = percentage value, result is 60.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/percentage-of-a-number/1%29%20Percentage%20of%20a%20Number%20Formula.webp)
  ### 🛎️ Finding a Percentage of an Amount
  - Turn the percentage into a **decimal or fraction**, whichever is easier
  - Example: 60% of 150 kg = 150 × 0.6 = 90 kg
  - ![Finding 60% of 150 kg equals 90 kg and 25% of 120 cm equals 30 cm using the formula base value × percentage rate = percentage value.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/percentage-of-a-number/2%29%20Percentage%20of%20a%20Number%20Examples.webp)
  ### 🛎️ Using Percentages in Real Life
  - First, find the **percentage value** (e.g. 20% of £180 = £36)
  - Then **add** it for a tip or tax, or **subtract** it for a discount
  - ![Calculating a 20% tip on a $180 bill, resulting in a $36 tip and a total payment of $216.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/percentage-of-a-number/3%29%20Percentage%20of%20a%20Number%20Application.webp)
  ### 🛎️ Percentage Tricks (10% and 1%)
  - **10%** of a number: move the decimal **one place left**
  - **1%** of a number: move the decimal **two places left**
  - ![Quick trick for finding percentage of a number and using it to calculate a 12% discount on a $1500 laptop.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/percentage-of-a-number/4%29%20Percentage%20of%20a%20Number%20Trick.webp)
- [Interest and Interest Rate Calculation](https://maths-angel.com/lessons/interest-rate-calculation)
  > Learn to calculate interest using Interest = Principal × Rate. Work through examples finding interest earned, the rate, and the principal. Watch free!
  ### 🛎️ Understanding Interest
  - **Interest** is the **extra money** earned or paid on an original amount.
  - The original amount is called the **principal**.
  - ![Understanding interest calculation with principal amount, interest rate, and total amount examples.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/interest-rate-calculation/1%29%20Interest%20and%20Interest%20Rate%20Calculation.webp)
  ### 🛎️ The Interest Formula
  - You use **interest = principal × interest rate** for **one year**.
  - If the time is shorter, you take the **fraction of a year**.
  - ![Example for Interest calculation, showing the formula for calculating interest on £200 at 3% over 60 days, resulting in a total amount of £201.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/interest-rate-calculation/2%29%20Calculating%20Interest.webp)
  ### 🛎️ Calculating the Interest Rate
  - You scale the **interest** to **one year** first.
  - You then **rearrange** to find **interest rate = interest ÷ principal**.
  - ![Example for calculating the interest rate, demonstrating the formula for interest with £200 principal and 4% interest rate.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/interest-rate-calculation/3%29%20Calculating%20Interest%20Rate.webp)
  ### 🛎️ Calculating the Principal
  - You **rearrange** the formula to make the **principal** the subject.
  - You divide the **interest** by the **interest rate** to find it.
  - ![Illustration of calculating the principal using a 5% interest rate, earning £30 after 1 year, with the formula and steps to solve for the principal.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/interest-rate-calculation/4%29%20Calculating%20Principal.webp)
- [Simple Interest and Compound Interest](https://maths-angel.com/lessons/simple-interest-and-compound-interest)
  > Simple interest earns a fixed amount each year. Compound interest grows faster by adding to the principal. Learn both formulas with clear examples. Watch free!
  ### 🛎️ How to Calculate Simple Interest?
  - **Interest = Principal × rate × time**, calculated on the **original amount** only.
  - The **total amount = Principal + Interest**, since the principal stays the same each year.
  - ![Simple interest calculation example, showing the principal amount, interest rate, and time in years, resulting in £15 interest over 3 years.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/simple-interest-and-compound-interest/1%29%20Simple%20Interest.webp)
  ### 🛎️ How to Calculate Compound Interest?
  - The **total amount** is calculated using **Total = Principal × (1 + rate)ᵗ**.
  - **Interest** is added to the **principal** each year, so the amount keeps increasing.
  - The **interest** is calculated using **Interest = Total − Principal**.
  - ![Compound interest calculation over three years showing the formula and total amount of £115.8 from an initial £100.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/simple-interest-and-compound-interest/2%29%20Compound%20Interest.webp)
  ### 🛎️ Simple and Compound Compared
  - **Simple interest** increases by the **same amount each year**.
  - **Compound interest** increases **faster over time** because interest is added to the total.
  - **Compound interest** is commonly used by banks.
  - ![Comparison of simple interest and compound interest formulas, showing calculations for total amount and interest earned over years.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/simple-interest-and-compound-interest/3%29%20Simple%20vs%20Compound%20Interest.webp)
- [Percentage Increase and Decrease](https://maths-angel.com/lessons/percentage-increase-and-decrease)
  > Learn how to calculate percentage increase and decrease using the formula: original × (1 ± change). Covers worked examples and reverse percentages. Watch free!
  ### 🛎️ The Percentage Formula
  - **New value = original value × (1 ± percentage change)**
  - Use “+” for an increase, and “−” for a decrease
  - ![Formula for percentage increase and decrease with examples showing 100 increasing by 20% and decreasing by 10%.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/percentage-increase-and-decrease/1%29%20Percentage%20Increase%20and%20Decrease%20-%20Formula.webp)
  ### 🛎️ Calculating Percentage Increase and Decrease
  - **Increase:** 60 increased by 20% = 60 × (1 + **20%**) = 60 × 1.2 = 72
  - **Decrease:** 90 with 30% off = 90 × (1 − **30%**) = 90 × 0.7 = 63
  - ![Percentage increase and decrease examples showing a score increase from 60 to 72 and a jacket price reduced from $90 to $63.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/percentage-increase-and-decrease/2%29%20Percentage%20Increase%20and%20Decrease%20-%20Examples.webp)
  ### 🛎️ Finding the Original Value (Reverse Percentage)
  - Use the **same formula**, but the original value is the **unknown**
  - Solve it by **dividing** (e.g. original × 0.6 = 36, so original = 36 ÷ 0.6 = 60)
  - ![Reverse percentage example showing how a $36 ticket with 40% discount gives an original price of $60.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/percentage-increase-and-decrease/3%29%20Reverse%20Percentage%20-%20Finding%20the%20Original%20Value.webp)
- [How to Calculate Percentage Change](https://maths-angel.com/lessons/percentage-change)
  > Percentage change shows how much a value increased or decreased. Use the formula (change ÷ original) × 100 with clear examples and exam tips. Watch free!
  ### 🛎️ How to Calculate Percentage Change?
  - **Percentage change** is found by dividing the **change** by the **original value** and multiplying by 100.
  - The **change** is the **new value minus the original value**.
  - ![Formula for calculating percentage change using change divided by original value times 100, with notes on increase and decrease.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/percentage-change/1%29%20How%20to%20Calculate%20Percentage%20Change.webp)
  ### 🛎️ Percentage Change Examples
  - If the final answer is **positive**, it is a **percentage increase**.
  - If the final answer is **negative**, it is a **percentage decrease**.
  - ![Percentage change examples showing a 30% price increase for a board game and a 25% price decrease for a clock.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/percentage-change/2%29%20Percentage%20Change%20Examples.webp)
  ### 🛎️ Practising Percentage Decrease
  - A **percentage decrease** happens when the new value is **smaller** than the original value.
  - The percentage shows how much the value has **decreased from the start**.
  - ![Percentage decrease calculation where a table bought for $1200 and sold for $960 shows a 20% drop over 3 years.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/percentage-change/3%29%20Percentage%20Change%20Practice.webp)
  ### 🛎️ Applying Percentage Change to Data
  - Always identify the **new value** and **original value** first before calculating.
  - **Exam tip:** always divide by the **original value**, not the new one.
  - ![Percentage change example showing a 5% decrease in cinema visitors from Week 2 (1500) to Week 4 (1425) using the change formula.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/percentage-change/4%29%20Percentage%20Change%20Application.webp)
- [Directly Proportional and Inversely Proportional](https://maths-angel.com/lessons/directly-proportional-inversely-proportional)
  > Learn the key difference between direct proportion (y = kx) and inverse proportion (xy = k). Spot which type applies with clear real-life examples. Watch free!
  ### 🛎️ What Is Direct Proportionality?
  - In **direct proportion**, doubling one value will **double** the other.
  - If 1 watermelon costs **£4**, then 2 cost **£8** and 4 cost **£16**.
  - ![Direct proportionality between the number of watermelons and their total cost, with calculations.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/directly-proportional-inversely-proportional/1%29%20Direct%20Proportion.webp)
  ### 🛎️ What Is Inverse Proportionality?
  - In **inverse proportion**, doubling one value will **halve** the other.
  - For a task, **15 people** take **2 hours**, but **30 people** only take **1 hour**.
  - ![Inverse proportion chart showing number of workers and hours needed, demonstrating that more workers reduce the time required to complete a task.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/directly-proportional-inversely-proportional/2%29%20Inverse%20Proportion.webp)
  ### 🛎️ Spotting the Difference Quickly
  - In **direct proportion**, the **unit rate** stays the same.
  - In **inverse proportion**, the **product** of the two quantities **stays the same**.
  - ![Direct and inverse proportionality concepts with real-life examples showing the relationship between quantities and costs or time.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/directly-proportional-inversely-proportional/3%29%20Direct%20Proportion%20vs%20Inverse%20Proportion.webp)
- [Direct Proportion Formula and Examples](https://maths-angel.com/lessons/direct-proportion)
  > Direct proportion means both values increase together using y = kx. Learn to find the constant k with examples including y ∝ x² and y ∝ √x. Watch free!
  ### 🛎️ What Is Direct Proportion?
  - Two quantities are in **direct proportion** if they **increase or decrease together**.
  - If **x doubles**, then **y also doubles**.
  - ![The definition of direct proportion with notation y∝x and formula y=kx (k is a constant).](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/direct-proportion/1%29%20What%20Is%20Direct%20Proportion.webp)
  ### 🛎️ Direct Proportion Formula
  - We use the formula **y = kx**, where **k ≠ 0**.
  - A **constant** means **k stays the same**, when x and y change.
  - ![Direct proportion example: y = kx with x=4, y=20 giving k=5, so y=5x and when x=10, y=50.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/direct-proportion/2%29%20Direct%20Proportion%20Formula%20Example%201.webp)
  ### 🛎️ Example: Direct Proportion to x²
  - If **y is directly proportional to x²**, we write **y = kx²**.
  - **Find k first** using given pairs, then **substitute the x value** to find the wanted y value.
  - ![Direct proportion to x² example: x=3, y=18 gives k=2, so y=2x² and when x=6, y=72](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/direct-proportion/3%29%20Direct%20Proportion%20Formula%20Example%202.webp)
  ### 🛎️ Example: Direct Proportion to √x
  - If **y is directly proportional to √x**, we write **y = k√x**.
  - **Find the constant k first**, then use the formula to **calculate the wanted y value**.
  - ![Direct proportion to √x example: x=49, y=14 gives k=2, so y=2√x and when x=100, y=20.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/direct-proportion/4%29%20Direct%20Proportion%20Formula%20Example%203.webp)
- [Inverse Proportion Formula and Examples](https://maths-angel.com/lessons/inverse-proportion)
  > Inverse proportion means one value increases as the other decreases, keeping xy = k constant. Learn the formula with examples for x, x², and √x. Watch free!
  ### 🛎️ What Is Inverse Proportion?
  - Inverse proportion means the **product stays constant**. For example, if x is doubled, y is halved, so xy stays the same.
  - In exams, **multiply the variables together** to get **the constant k**. For example, if x = 3 and y = 8, then k = 3 × 8 = 24.
  - ![Inverse proportion with a definition, notation showing y is proportional to 1 divided by x, and the formula y = k divided by x.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/inverse-proportion/1%29%20What%20is%20Inverse%20Proportion.webp)
  ### 🛎️ Example: Inverse Proportion to x
  - If y is **inversely proportional to x**, write **xy = k**.
  - **Multiply** the given pair to **find k** first, then **substitute the x value** to find the wanted y value.
  - ![Inverse proportion example y ∝ 1/x: x=6 gives y=8, so k=48 and y=12 when x=4.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/inverse-proportion/2%29%20Inverse%20Proportion%20Formula%20Example%201.webp)
  ### 🛎️ Example: Inverse Proportion to x²
  - If y is **inversely proportional to x²**, write **x²y = k**.
  - **Multiply y by x²** to find **k**, then **substitute the x value** to find the wanted y value.
  - ![Worked example of inverse proportion y ∝ 1/x²: x=3 gives y=8, so k=72 and y=2 when x=6.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/inverse-proportion/3%29%20Inverse%20Proportion%20Formula%20Example%202.webp)
  ### 🛎️ Example: Inverse Proportion to √x
  - If y is **inversely proportional to √x**, write **√x · y = k**.
  - **Multiply √x by y** to find **k**, then **substitute the x value** to find the wanted y value.
  - ![Inverse proportion example n ∝ 1/√r: n=12 when r=9, so k=36 and n=9 when r=16.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/ratio-proportion-rates-of-change/inverse-proportion/4%29%20Inverse%20Proportion%20Formula%20Example%203.webp)

## Geometry & Measures

- [Perpendicular and Parallel Lines](https://maths-angel.com/lessons/perpendicular-and-parallel-lines)
  > Perpendicular lines meet at 90° and parallel lines never intersect. Learn to identify line types and measure distances between points and lines. Watch free!
  ### 🛎️ What are Different Types of Lines?
  - A **line segment** has two endpoints, a **ray** has one endpoint, and a **line** has no endpoints.
  - Rays and lines continue forever, but a line segment has a **fixed length**.
  - ![Definition, notation, and visualisation of line segments, rays, and lines.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/perpendicular-and-parallel-lines/1%29%20Segments%2C%20Rays%2C%20Lines.webp)
  ### 🛎️ What are Perpendicular and Parallel Lines?
  - **Perpendicular lines** meet at a **right angle (90°)**.
  - **Parallel lines** stay the **same distance apart** and **never meet**.
  - ![Characteristics, notation, and visualisation of perpendicular and parallel lines.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/perpendicular-and-parallel-lines/2%29%20Perpendicular%20vs%20Parallel%20Lines.webp)
  ### 🛎️ How to Measure Distance Between Points and Lines
  - The distance between **two points** is the length of the **line segment** joining them.
  - The distance between **parallel lines** is the length of the **perpendicular line segment** between them.
  - ![Measuring distances between points and lines, showing examples of point to point, point to line, and between parallel lines.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/perpendicular-and-parallel-lines/3%29%20Measuring%20Distances.webp)
- [Measurement and Types of Angles](https://maths-angel.com/lessons/measurement-and-types-of-angles)
  > Angles form where two lines meet at a vertex. Learn about acute, right, obtuse, and reflex angles, and how to measure them with a protractor. Watch free!
  ### 🛎️ What is an Angle?
  - An angle is formed when **two lines meet at a point**, called the **vertex**.
  - Angles are measured in **degrees (°)**.
  - ![Definition of an angle: an angle is formed when two lines meet at a point (the vertex), opens anticlockwise, and is measured in degrees.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/measurement-and-types-of-angles/1%29%20What%20is%20an%20angle.webp)
  ### 🛎️ Types of Angles
  - **Acute angle:** 0° < angle < 90°
  - **Obtuse angle:** 90° < angle < 180°
  - **Reflex angle:** 180° < angle < 360°
  - ![Illustration of acute, right, obtuse, straight, reflex, and full angles, with corresponding angle measures.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/measurement-and-types-of-angles/2%29%20Categorising%20Angles.webp)
  ### 🛎️ How to Measure Angles?
  - Place the **centre of the protractor** on the **vertex**.
  - Align the **zero line** with one arm and **read where the other arm** crosses the scale.
  - ![Steps for measuring angles using a protractor, with instructions and an illustration of a protractor aligned to measure an angle of 130 degrees.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/measurement-and-types-of-angles/3%29%20Measuring%20Angles.webp)
  ### 🛎️ Measuring Angles Greater Than 180°
  - Measure the **smaller angle first** using the protractor.
  - **Subtract from 360°** to find the reflex angle.
  - ![Protractor measuring a reflex angle with calculation showing 360 degrees minus 140 degrees equals 220 degrees.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/measurement-and-types-of-angles/4%29%20Measuring%20Angles%20over%20180%20degrees.webp)
  ### 🛎️ How to Draw Angles?
  - Draw a **straight line** for one arm of the angle.
  - Place the protractor, **mark the required angle,** then draw the second arm.
  - ![Steps for drawing a 40-degree angle using a protractor.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/measurement-and-types-of-angles/5%29%20Drawing%20Angles.webp)
  ### 🛎️ Drawing Angles Greater Than 180°
  - **Subtract the given angle from 360°** to find the smaller angle.
  - Draw the **smaller angle**, and the **reflex angle** is the one you need.
  - ![Drawing a 210-degree reflex angle using a protractor with a pencil, and a diagram showing the sum of 210 and 150 degrees equals 360 degrees.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/measurement-and-types-of-angles/6%29%20Drawing%20Angles%20over%20180%20degrees.webp)
- [Angle Relationships in Intersecting and Parallel Lines](https://maths-angel.com/lessons/angle-relationships)
  > Angle relationships form when a line crosses two parallel lines. Learn corresponding, alternate, and co-interior angles with clear examples. Watch free!
  ### 🛎️ Angles Formed by Intersecting Lines
  - **Vertically opposite angles** are equal
  - **Adjacent angles** on a straight line add to **180°**
  - ![Illustrating that vertically opposite angles are equal, and adjacent angles sum to 180° for two intersecting lines.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/angle-relationships/1%29%20Vertically%20Opposite%20and%20Adjacent%20Angles.webp)
  ### 🛎️ Angles in Parallel Lines with a Transversal
  - **Corresponding angles** are equal
  - **Alternate angles** are equal
  - **Co-interior angles** add to **180°**
  - ![Diagram showing corresponding angles, alternate angles, and co-interior angles formed by two parallel lines and a transversal.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/angle-relationships/2%29%20Corresponding%2C%20Alternate%2C%20Co-Interior%20Angles.webp)
  ### 🛎️ How to Find Missing Angles
  - Use **equal angles** (vertically opposite, corresponding, alternate)
  - Use **angles that add to 180°** (co-interior angles, supplementary angles)
  - ![Calculations of vertically opposite, supplementary, alternate, and corresponding angles with given values of 40° and 140°.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/angle-relationships/3%29%20Finding%20the%20Missing%20Angles.webp)
- [Polygons and Types of Quadrilaterals](https://maths-angel.com/lessons/polygons-quadrilaterals)
  > Polygons are 2D shapes with straight sides. Learn quadrilateral types like rhombus and parallelogram, plus the interior angles formula. Watch free!
  ### 🛎️ What is a Polygon?
  - A **polygon** is a **2D closed shape** made from **straight line segments**.
  - Tip: shapes that are open, curved, or 3D are **not** polygons.
  - ![Illustrating polygons as 2D closed shapes with straight line segments, showing a triangle and star, and non-polygons like curved shape and 3D cube.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/polygons-quadrilaterals/1%29%20Polygons%20vs%20non-polygons.webp)
  ### 🛎️ Names of Polygons
  - Polygons are named by their **number of sides**.
  - For example: **triangle (3), quadrilateral (4), pentagon (5), hexagon (6)**.
  - ![Table showing polygons with their corresponding number of sides.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/polygons-quadrilaterals/2%29%20Regular%20Polygons.webp)
  ### 🛎️ Interior Angles of Polygons
  -	The sum of interior angles depends on the number of sides.
  - The formula is **(n − 2) × 180°**, where n is the **number of sides**.
  - ![Table showing polygons, their number of sides, and sum of interior angles. Triangle (180°), quadrilateral (360°), pentagon (540°), and hexagon (720°).](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/polygons-quadrilaterals/3%29%20Interior%20Angles%20of%20Polygons.webp)
  ### 🛎️ What is a Quadrilateral?
  -	A quadrilateral is a polygon with four sides.
  - **Parallelograms, rectangles, rhombuses, squares, and trapeziums** are all quadrilaterals.
  - ![Illustrating special types of quadrilaterals, highlighting their unique properties such as parallel sides and equal angles.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/polygons-quadrilaterals/4%29%20Special%20Quadrilaterals.webp)
- [Properties of Circles](https://maths-angel.com/lessons/properties-of-circles)
  > All points on a circle are equidistant from the centre. Learn about radius, diameter, chords, sectors, and tangents with clear definitions. Watch free!
  ### 🛎️ Properties of a Circle
  - The **diameter** goes through the **centre** and is **twice the radius**.
  - A **sector** is made from **two radii** and the **arc** between them.
  - A **tangent** touches the circle at **one point** and is **perpendicular to the radius**.
  - ![Diagram of a circle showing key terms: tangent, chord, diameter, radius, sector, segment, and arc, with definitions around the circle.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/properties-of-circles/1%29%20Circles%20and%20Key%20Terms.webp)
  ### 🛎️ Finding Points with Circles
  - All points the same distance from a point lie on a **circle**, and that distance is the **radius**.
  - The **intersections** of circles show the **possible answers**.
  - ![Diagram illustrating how to find points that are 5 cm from point A and 3 cm from point B, with circles drawn for each point to show intersections.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/properties-of-circles/2%29%20Finding%20Points%20with%20Circles.webp)
- [Thales' Theorem](https://maths-angel.com/lessons/thales-theorem)
  > Thales' Theorem states that any triangle drawn on a semicircle's diameter has a 90° angle. Learn to find missing angles with this rule. Watch free!
  ### 🛎️ Thales’ Theorem: What It Says
  - If one side of a **triangle** is the **diameter** of a circle, the opposite angle is **90°**.
  - This angle is always a **right angle**, wherever the point is on the **circle**.
  - ![Explanation of Thales's Theorem with a diagram showing a right-angled triangle formed by a diameter and any point on a semicircle.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/thales-theorem/1%29%20What%20is%20Thales%27%20Theorem.webp)
  ### 🛎️ Thales’ Theorem: How to Use It
  - First, find the **diameter** of the circle.
  - Then mark the angle opposite the **diameter** on the circle as **90°**.
  - ![Diagram for Thales' theorem with triangle inscribed in a semicircle, showing a right angle at point C and interior angles A = 35° and B = 55°.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/thales-theorem/2%29%20How%20to%20Apply%20Thales%27%20Theorem.webp)
  ### 🛎️ Thales’ Theorem: Finding Other Angles
  - All angles in a **triangle** add up to **180°**.
  - Subtract **90°** and the given angle to find the **missing angle**.
  - ![Semicircle with diameter AB, triangle ACB showing angle ACB as 90° using Thales's theorem, with marked angles 40°, 50°, and 90°.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/thales-theorem/3%29%20Thales%27%20Theorem%20Exercise.webp)
- [Line of Symmetry and Reflection Symmetry](https://maths-angel.com/lessons/line-of-symmetry)
  > A line of symmetry divides a shape into two identical halves. Learn to count symmetry lines in common shapes and spot reflection symmetry. Watch free!
  ### 🛎️ What is a Line of Symmetry?
  - A line of symmetry divides a shape into **two identical halves**.
  - If you **fold the shape along the line**, both sides match exactly.
  - ![Heart shape divided by a line of symmetry, showing that folding along the line makes both halves match perfectly.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/line-of-symmetry/1%29%20What%20is%20Line%20of%20Symmetry.webp)
  ### 🛎️ Lines of Symmetry in Common Shapes
  - The number of lines depends on how many ways a shape can **fold exactly in half**.
  - Squares have **4**, rectangles have **2**, and circles have **infinite** lines.
  - ![Lines of symmetry in shapes: rectangle 2, square 4, isosceles triangle 1, equilateral triangle 3, circle infinite.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/line-of-symmetry/2%29%20Line%20of%20Symmetry%20in%20Common%20Shapes.webp)
  ### 🛎️ How to Check for Reflection Symmetry?
  - Try folding the shape along a line and see if **both sides match exactly**.
  - If you can find **at least one line of symmetry**, the shape has **reflection symmetry**.
  - ![Four road signs are checked for whether they have reflection symmetry.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/line-of-symmetry/3%29%20Check%20for%20Reflection%20Symmetry.webp)
  ### 🛎️ Key Features of a Line of Symmetry
  - Every point on one side has a **mirror point** on the other side.
  - Mirror points are the **same distance** from the line of symmetry.
  - ![Diagram showing a pentagon with a vertical line of symmetry and demonstrating two key features of a line of symmetry.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/line-of-symmetry/4%29%20Key%20Feature%20of%20Line%20of%20Symmetry.webp)
  ### 🛎️ Completing a Shape Given Line of Symmetry
  - **Measure the distance** from a point to the line of symmetry.
  - Plot the mirror point the **same distance** on the **other side**.
  - **Connect the points** to complete the shape.
  - ![Step-by-step guide to completing a shape using a line of symmetry: measure, locate mirror points, and connect them.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/line-of-symmetry/5%29%20Complete%20a%20Shape%20Given%20Line%20of%20Symmetry.webp)
- [Rotational Symmetry](https://maths-angel.com/lessons/rotational-symmetry)
  > A shape has rotational symmetry if it looks the same after turning less than 360°. Learn the centre, angle, and order of rotational symmetry with examples. Watch free!
  ### 🛎️ What is Rotational Symmetry?
  - A shape has **rotational symmetry** if it looks the same **after a turn of less than 360°**.
  - The shape is rotated about its **centre of rotation**.
  - ![Diagram showing a square with an order of rotation symmetry 4, and explanations of the centre of rotation, angle of rotation, and order of symmetry.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/rotational-symmetry/1%29%20Rotational%20Symmetry.webp)
  ### 🛎️ Order and Angle of Rotation
  - The **order of symmetry** is the number of times a shape matches itself in a full 360° turn.
  - The **angle of rotation** = **360° ÷ order** (e.g. a rectangle has order 2, so the angle is 180°).
  - ![Rotational symmetry of rectangle, equilateral triangle, and regular hexagon, showing their centre of rotation, angle of rotation, order of symmetry.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/rotational-symmetry/2%29%20Rotational%20Symmetry%20of%20Common%20Shapes.webp)
  ### 🛎️ How to Check for Rotational Symmetry?
  -	Rotate the shape around its centre and **see if it matches its starting position**.
  -	If it **matches before 360°**, the shape has rotational symmetry.
  - ![Illustrating three traffic signs to check for whether they have rotational symmetry, and showing different angles of rotation and orders of symmetry.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/rotational-symmetry/3%29%20Check%20for%20Rotational%20Symmetry.webp)
- [Plotting and Reflecting Points on the Coordinate Plane](https://maths-angel.com/lessons/reflecting-points-coordinate-plane)
  > Reflecting a point across the x-axis, y-axis, or origin flips its coordinates on the coordinate plane. Learn the rules with plotted examples. Watch free!
  ### 🛎️ The Coordinate System
  - The **x-axis** is horizontal and the **y-axis** is vertical.
  - The point where they meet is called the **origin** (0,0).
  - ![Coordinate system graph showing positive and negative numbers, with the origin marked at (0,0).](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/reflecting-points-coordinate-plane/1%29%20Coordinate%20System.webp)
  ### 🛎️ How to Locate a Point on a Grid
  - A point is written as **(x, y)**.
  - Move **along x first**, then **up or down y**.
  - ![Locating points on a coordinate system with positive and negative coordinates.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/reflecting-points-coordinate-plane/2%29%20Locating%20Points%20on%20Coordinate%20System.webp)
  ### 🛎️ Reflecting Points over y-axis
  - Change the **sign of the x-coordinate**.
  - The **y-coordinate stays the same**.
  - ![Explanation on how to reflect (-3, 2) on a coordinate grid across the y-axis to (3, 2), by changing the sign of x.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/reflecting-points-coordinate-plane/3%29%20Reflecting%20Points%20over%20the%20y-axis.webp)
  ### 🛎️ Reflecting Points over x-axis
  - Change the **sign of the y-coordinate**.
  - The **x-coordinate stays the same**.
  - ![Explanation on how to reflect (-3, 2) on a coordinate grid across the x-axis to (-3, -2), by changing the sign of y.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/reflecting-points-coordinate-plane/4%29%20Reflecting%20point%20over%20x-axis.webp)
  ### 🛎️ Reflecting in the Origin
  - Change the **sign of both x and y**.
  - The point moves to the **opposite quadrant**.
  - ![Explanation on how to reflect (-3, 2) on a coordinate grid across the origin to (3, -2), by changing both the signs of x and y.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/reflecting-points-coordinate-plane/5%29%20Reflecting%20Point%20over%20the%20origin.webp)
  ### 🛎️ Reflecting Shapes
  - Reflect **every vertex** using the same rule.
  - Then **connect the reflected points** to form the new shape.
  - ![Reflecting a triangle over the y-axis by changing the sign of the x-coordinate of all points.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/reflecting-points-coordinate-plane/6%29%20Reflecting%20Figure%20over%20y-axis.webp)
- [Translating Shapes](https://maths-angel.com/lessons/translating-shapes)
  > Translation slides a shape on a grid without changing size or orientation. Learn to use column vectors to describe and apply translations. Watch free!
  ### 🛎️ Translation: Sliding a Shape
  - A **translation** moves a shape without changing its **size** or **orientation**.
  - Every point shifts the **same distance** in the **same direction**.
  - ![Column vectors with visual examples. Describes horizontal and vertical shifts using positive and negative values.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/translating-shapes/1%29%20Translation.webp)
  ### 🛎️ Column Vectors
  - The **top number** shows the **horizontal movement**, where positive is right and negative is left.
  - The **bottom number** shows the **vertical movement**, where positive is up and negative is down.
  - ![Examples of vectors as ordered pairs, illustrating horizontal and vertical shifts on a grid with arrows and numerical values.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/translating-shapes/2%29%20Column%20vectors.webp)
  ### 🛎️ Translating 2D Shapes
  - You move all **vertices** using the **same column vector**, then connect them.
  - The shape stays **identical**, just in a **new position**.
  - ![Shape P is translated to shape Q on a grid by column vector (8, -3), with horizontal shift of 8 units and vertical shift of -3 units.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/translating-shapes/3%29%20Translating%20with%20a%20column%20vector.webp)
  ### 🛎️ Finding the Column Vector
  - Count how far a point moves **left or right** first.
  - Then count how far it moves **up or down**.
  - ![Translation of triangle M to triangle N on a grid using column vector (-10, -4), with -10 units left and -4 units down.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/translating-shapes/4%29%20Find%20the%20column%20vector%20of%20a%20translation.webp)
- [Rotating Shapes](https://maths-angel.com/lessons/rotating-shapes)
  > Rotation turns a shape around a fixed point called the centre of rotation. Learn to rotate shapes using tracing paper and drawing methods. Watch free!
  ### 🛎️ What Is Rotation?
  - A **rotation** turns a shape around a **fixed point** called the **centre**.
  - The shape keeps the **same size and shape**, only its **position changes**.
  - ![Diagram explaining rotation in geometry with examples, including the centre of rotation, direction (clockwise or anticlockwise), and angle.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/rotating-shapes/1%29%20What%20is%20Rotation.webp)
  ### 🛎️ Rotating Shapes Using Tracing Paper
  - Place **tracing paper** on the shape and **mark the centre**.
  - Turn the paper by the given **angle** in the correct **direction**.
  - ![Rotating shape P 90 degrees clockwise about point O on a coordinate grid, resulting in shape Q.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/rotating-shapes/2%29%20Rotate%20Shapes%20with%20a%20Tracing%20Paper.webp)
  ### 🛎️ Rotating Shapes on a Grid
  - Draw a line from the **centre of rotation** to one **vertex**.
  - Turn the line by the **given angle**, keeping the **same distance from the centre**.
  - Repeat for all vertices and **join the points** to form the new shape.
  - ![Rotating shape P 90° anticlockwise about point O using auxiliary lines, resulting in shape Q, shown on a grid with axes.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/rotating-shapes/3%29%20Rotate%20Shapes%20by%20Drawing%20Lines.webp)
  ### 🛎️ Describing a Rotation
  - Draw **perpendicular bisectors** of matching points to find the **centre of rotation**.
  - Measure the **angle at the centre** from the **original point to the rotated point**.
  - ![Rotating  triangle P rotated 90° anticlockwise about the (2, 3) to triangle Q, with perpendicular bisectors intersecting at the centre of rotation.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/rotating-shapes/4%29%20How%20to%20Find%20the%20Centre%20of%20Rotation.webp)
- [Introduction to Enlargement](https://maths-angel.com/lessons/enlargement)
  > Enlargement changes a shape's size using a scale factor from a fixed centre. Learn to enlarge shapes and find the scale factor with examples. Watch free!
  ### 🛎️ What Is Enlargement?
  - **Enlargement** is a **transformation** that changes the **size** of a shape.
  - All **angles stay the same** after an enlargement, but the lengths **get bigger or smaller**.
  - ![Illustration explaining enlargement as a transformation, showing triangles scaled from widths 2 to 4 to 8.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/enlargement/1%29%20What%20Is%20Enlargement.webp)
  ### 🛎️ Understanding the Scale Factor
  - The **scale factor** is the number you multiply all lengths by.
  - If the scale factor is **greater than 1**, the shape gets **bigger**, and if it is **between 0 and 1**, the shape gets **smaller**.
  - ![Enlarged triangles on a coordinate grid from centre P, showing scale factor 2, 4 and 8 and that factors >1 enlarge and 0-1 reduce](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/enlargement/2%29%20Key%20Concepts%20in%20Enlargement.webp)
  ### 🛎️ How to Enlarge a Shape from a Centre
  - Draw straight **lines from the centre of enlargement** to each corner.
  - Measure the distance from the centre to a corner, then mark the new point **3 times as far** from the centre.
  - ![Triangle enlarged by scale factor 3 from point P on a coordinate grid, with rays from P to vertices and new vertices marked three times further away](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/enlargement/3%29%20How%20to%20Enlarge%20a%20Shape.webp)
  ### 🛎️ Finding the Centre of Enlargement
  - Join each corner of the original shape to the matching corner of the enlarged shape.
  - **Extend the lines** until they meet at one point, called the **centre of enlargement**.
  - ![Coordinate grid showing trapezium A enlarged to trapezium B with lines from matching vertices meeting at centre of enlargement (0,0).](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/enlargement/4%29%20Finding%20the%20Centre%20of%20Enlargement.webp)
  ### 🛎️ Finding the Scale Factor
  - The **scale factor** is found using new length ÷ original length.
  - If a side goes from **4 units to 8 units**, then 8 ÷ 4 = 2, so the scale factor is **2**.
  - **Exam tip:** Choose **any pair of matching sides**. **Horizontal or vertical lines** on the grid are easiest.
  - ![Trapezium A enlarged to trapezium B on a coordinate grid, with lines to centre of enlargement O and scale factor 2 shown by lengths 4 and 8](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/enlargement/5%29%20Enlargement%20from%20Trapezium%20A%20to%20B.webp)
- [Constructing Triangles](https://maths-angel.com/lessons/constructing-triangles)
  > Constructing triangles uses specific combinations of sides and angles. Learn ASA, SAS, and SSS methods with a compass and protractor. Watch free!
  ### 🛎️ Parts of a Triangle Explained
  - The **vertices** A, B and C are labelled in an **anticlockwise** direction.
  - Each **side** a, b and c is **opposite** its matching **angle**.
  - ![A triangle with vertices A, B, and C, sides a, b, and c, and angles alpha, beta, and gamma. The vertices are labelled anticlockwise.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/constructing-triangles/1%29%20Triangle%20Components.webp)
  ### 🛎️ Constructing a Triangle with Two Angles and the Included Side (ASA)
  - The given **side** is the **side between** the two known **angles**.
  - Draw the **side first**, then measure each **angle** at the ends so they meet at **C**.
  - ![Constructing a triangle ABC with angles 60° and 70°, side c = 6 cm using compass and ruler. The sides and angles are labelled.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/constructing-triangles/2%29%20Construct%20Triangle%20(ASA%29.webp)
  ### 🛎️ Constructing a Triangle with Two Sides and the Included Angle (SAS)
  - Draw one **side**, then use a **protractor** to draw the **angle at one end**.
  - Measure the second **side** along the angle to find point **C**.
  - ![Constructing a triangle ABC with sides 5 cm and 6 cm, and angle 50°, including labelled diagram.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/constructing-triangles/3%29%20Construct%20Triangle%20(SAS%29.webp)
  ### 🛎️ Common Pitfall: Two Possible Triangles
  - With two **sides** and a **non-included angle**, there may be **two triangles**.
  - This is called the **ambiguous case** and both triangles can be correct.
  - ![Constructing a triangle ABC with sides a = 7 cm, b = 5 cm, and angle beta = 40°, showing two possible triangles with different angles at vertex C.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/constructing-triangles/4%29%20Constructing%20Triangles(Common%20Pitfall%29.webp)
  ### 🛎️ Constructing a Triangle with Three Sides (SSS)
  - Draw the **longest side** first to make construction easier.
  - Use a **compass** to draw **arcs** from each end that meet at point **C**.
  - ![Constructing triangle ABC with sides 4 cm, 5 cm, and 6 cm using compass and ruler, including labelled angles alpha, beta, and gamma.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/constructing-triangles/5%29%20Construct%20Triangle%20(SSS%29.webp)
- [Congruent Triangles](https://maths-angel.com/lessons/congruent-triangles)
  > Congruent triangles have equal sides and angles. Learn the four rules of congruence — SSS, SAS, ASA, and RHS — with clear examples and visuals. Watch free!
  ### 🛎️ What Are Congruent Triangles?
  - Two triangles are **congruent** if they have the **same size and shape**.
  - This means **all corresponding sides and angles are equal**.
  - ![Diagram showing the criteria for identifying congruent triangles: SSS, SAS, ASA, and RHS with corresponding labelled triangles.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/congruent-triangles/1%29%20Congruent%20Triangles%20Definition.webp)
  ### 🛎️ How Can You Recognise Congruent Triangles?
  - Congruent triangles can be moved to **overlap perfectly**.
  - This can be done by **translating, reflecting, or rotating**.
  - ![Diagram showing the criteria for identifying congruent triangles: SSS, SAS, ASA, and RHS with corresponding labelled triangles.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/congruent-triangles/2%29%20Congruent%20Triangles%20Rules.webp)
  ### 🛎️ SSS (Side-Side-Side)
  - Two triangles are congruent if **all three corresponding sides are equal**.
  - The angles do **not** need to be given.
  - ![Identifying congruent triangles using SSS (side, side, side) criteria with two triangles both having sides of 3 cm, 5 cm, and 6 cm.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/congruent-triangles/3%29%20Congruent%20Triangles%20(Side-Side-Side%29.webp)
  ### 🛎️ SAS (Side-Angle-Side)
  - Two triangles are congruent if **two corresponding sides and the included angle are equal**.
  - The angle must be the **included angle**.
  - ![Identifying congruent triangles using SAS (Side, Angle, Side) rule with two triangles having sides of 3 cm, 5 cm, and an included angle of 70 degrees.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/congruent-triangles/4%29%20Congruent%20Triangles%20(Side-Angle-Side%29.webp)
  ### 🛎️ ASA (Angle-Side-Angle)
  - Two triangles are congruent if **two corresponding angles and the included side are equal**.
  - The side must be **between the two angles**.
  - ![Identifying congruent triangles using ASA (Angle, Side, Angle) with two example triangles showing two angles and the included side as the same.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/congruent-triangles/5%29%20Congruent%20Triangles%20(Angle-Side-Angle%29.webp)
  ### 🛎️ RHS (Right-Angle-Hypotenuse-Side)
  - Applies **only to right-angled triangles**.
  - Triangles are congruent if they have the **same hypotenuse and one equal side**.
  - ![Identifying congruent triangles using RHS criteria with examples of right-angle triangles having the same hypotenuse and one identical side.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/congruent-triangles/6%29%20Congruent%20Triangles%20(RHS%29.webp)
  ### 🛎️ Common Pitfall
  - SSA does **NOT** guarantee congruence.
  - Two equal sides and a **non-included angle** are **not enough** to prove congruence.
  - ![Illustrating that two triangles are not necessarily congruent if they share two equal sides and a non-included angle.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/congruent-triangles/7%29%20Congruent%20Triangles%20(Common%20Pitfall%29.webp)
- [Perpendicular Bisectors and Circumcircle](https://maths-angel.com/lessons/perpendicular-bisectors-and-circumcircle)
  > A perpendicular bisector divides a line segment in half at 90°. Learn how to construct them and find a triangle's circumcentre and circumcircle. Watch free!
  ### 🛎️ What Is a Perpendicular Bisector?
  - A **perpendicular bisector** cuts a **line segment in half**.
  - It meets the segment at a **right angle (90°)** at the **midpoint**.
  - ![Perpendicular bisector dividing a line segment at a 90-degree angle with midpoint marked.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/perpendicular-bisectors-and-circumcircle/1%29%20Perpendicular%20Bisector.webp)
  ### 🛎️ Drawing a Perpendicular Bisector
  - With a **compass**, set the width **greater than half** the segment.
  - Draw arcs **above and below** from **both ends**, keeping the **same width**.
  - Draw a straight line through the **two points where the arcs cross**.
  - ![Instructions for drawing a perpendicular bisector with a ruler and a compass.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/perpendicular-bisectors-and-circumcircle/2%29%20Drawing%20a%20Perpendicular%20Bisector.webp)
  ### 🛎️ Key Property of a Perpendicular Bisector
  - Every point on the **perpendicular bisector** is the **same distance** from **both ends** of the line segment.
  - This helps you find points that are **exactly in the middle** between the two ends.
  - ![Triangle with perpendicular bisectors meeting at the circumcentre, inside a circumcircle passing through all three vertices.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/perpendicular-bisectors-and-circumcircle/3%29%20Key%20Property%20of%20the%20Perpendicular%20Bisector.webp)
  ### 🛎️ Perpendicular Bisectors and the Circumcentre
  - The **circumcentre** is where the three **perpendicular bisectors** of a triangle meet.
  - It is **equal distance from all three vertices** and is the centre of the **circumcircle**.
  - ![Diagram of a triangle’s circumcentre, showing the circumradius, circumscribed circle (circumcircle), and the intersection of perpendicular bisectors.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/perpendicular-bisectors-and-circumcircle/4%29%20Triangle%20Circumcentre.webp)
- [Angle Bisectors and Incircle](https://maths-angel.com/lessons/angle-bisectors-and-incircle)
  > Angle bisectors divide an angle into two equal parts. Learn how to construct them, find a triangle's incentre, and draw the incircle with examples. Watch free!
  ### 🛎️ What Is an Angle Bisector?
  - An **angle bisector** splits an **angle** into **two equal parts**.
  - Each new angle is exactly **half** of the original angle.
  - ![Angle bisector dividing an angle into two equal parts, with one half labelled alpha over two and the other half also labelled alpha over two.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/angle-bisectors-and-incircle/1%29%20Angle%20Bisector.webp)
  ### 🛎️ How to Draw an Angle Bisector?
  - You can draw an **angle bisector** using a **protractor** by halving the angle.
  - You can also use a **compass** to draw the bisector without measuring.
  - ![Constructing an angle bisector using a protractor and compass, showing steps for accurate measurement and drawing.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/angle-bisectors-and-incircle/2%29%20Drawing%20Angle%20Bisectors.webp)
  ### 🛎️ The Key Property of an Angle Bisector
  - Any point on an **angle bisector** is **equidistant** from both sides of the angle.
  - This means the **perpendicular distance** to each side is the same.
  - ![Key property of angle bisectors shown with diagram: every point on the angle bisector is equidistant from the angle's sides.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/angle-bisectors-and-incircle/3%29%20Angle%20Bisector%20Property.webp)
  ### 🛎️ Incentre and Incircle
  - The **incentre** is where the **three angle bisectors** of a triangle meet.
  - The **incircle** is the **largest circle** that fits inside the triangle and touches all **three sides**.
  - ![Explanation of a triangle's in-centre and in-circle, showing the largest circle that can fit inside the triangle, with in-radius and angle bisectors.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/angle-bisectors-and-incircle/4%29%20Triangle%20Incircle.webp)
- [Medians and Centroid of a Triangle](https://maths-angel.com/lessons/medians-and-centroid-of-a-triangle)
  > The medians of a triangle connect each vertex to the opposite midpoint and meet at the centroid. Learn the 2:1 ratio with clear examples. Watch free!
  ### 🛎️ Median of a Triangle
  - A **median** joins a **vertex** to the **midpoint** of the opposite side.
  - A median splits a triangle into **two smaller triangles** with **equal area**.
  - ![The definition of a median of a triangle is that it connects a vertex to the midpoint of the opposite side, dividing it into two equal-area triangles.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/medians-and-centroid-of-a-triangle/1%29%20Median%20of%20a%20Triangle.webp)
  ### 🛎️ Centroid of a Triangle
  - The **centroid** is the point where the **three medians** meet.
  - It divides each **median** in the ratio **2 : 1**, measured from the **vertex**.
  - ![The definition of the centroid of a triangle is the point where the three medians intersect, dividing each median in a 2:1 ratio.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/medians-and-centroid-of-a-triangle/2%29%20Centroid%20of%20a%20Triangle.webp)
  ### 🛎️ Using Medians and the Centroid
  - A **median** always goes to the **midpoint** of the opposite side.
  - The **centroid** splits each median in a **2 : 1 ratio**, with the longer part next to the **vertex**.
  - ![The median and centroid of a triangle apply to the problem.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/medians-and-centroid-of-a-triangle/3%29%20Median%20and%20Centroid%20of%20a%20Triangle.webp)
- [Units of Area](https://maths-angel.com/lessons/units-of-area)
  > Area measures how much space a surface covers, in units like mm², cm², and m². Learn to convert between units of area with clear examples. Watch free!
  ### 🛎️ What is Area?
  - Area measures **how much space a flat shape covers.**
  - It is measured using **square units.**
  - ![Explaining the concept of area using 25 unit squares, each 1 square metre, to calculate a total area of 25 square metres with a room diagram.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/units-of-area/1%29%20What%20is%20Area.webp)
  ### 🛎️ Common Units of Area
  - Common units include **mm², cm², m², and km².**
  - Each unit represents a **square** (e.g. 1 cm² = 1 cm × 1 cm).
  - ![Table comparing units of area: square millimetre, square centimetre, square metre, and square kilometre with examples](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/units-of-area/2%29%20Common%20Units%20of%20Area.webp)
  ### 🛎️ Converting Units of Area
  - When converting area, you must **square the length conversion.**
  - For example, 1 cm = 10 mm, so **1 cm² = (10 mm)² = 100 mm².**
  - ![Conversion for units of area showing the relationships between square millimetres, centimetres, metres, and kilometres, with examples and steps.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/units-of-area/3%29%20Converting%20Units%20of%20Area%20(Chart%29.webp)
  ### 🛎️ Using Area Conversions
  - When converting to a **smaller area unit** (e.g. m² → cm²), **multiply.**
  - When converting to a **larger area unit** (e.g. mm² → cm²), **divide.**
  - ![Explaining area unit conversions, including mm², cm², m², and km², with examples for converting 5000 mm² to 50 cm² and 3 m² to 30,000 cm².](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/units-of-area/4%29%20Converting%20Area%20Units%20(Examples%29.webp)
- [Area of a Triangle](https://maths-angel.com/lessons/area-of-a-triangle)
  > The area of a triangle is calculated using ½ × base × height. Explore step-by-step examples for regular and right-angled triangles. Watch free!
  ### 🛎️ What is the Area of a Triangle
  - The area of a triangle is given by: Area = 1/2 × base × height
  - The **height must be perpendicular** to the base.
  - ![Diagram explaining the area of triangles formula  1/2 x b x h, examples of perpendicular height, and notes emphasising any side as base.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/area-of-a-triangle/1%29%20Area%20of%20Triangles%20Formula.webp)
  ### 🛎️ Finding the Area of a Triangle
  - If you know a **base** and its **perpendicular height**, you can find the area.
  - Multiply **base × height**, then **divide by 2**.
  - ![Illustration of finding the area of a triangle with a base of 6 cm and height of 5 cm using the formula 1/2 x base x height, resulting in 15 cm².](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/area-of-a-triangle/2%29%20Regular%20triangles%20Example.webp)
  ### 🛎️ Area of a Right-Angled Triangle
  -	In a right-angled triangle, the two **perpendicular sides** are called the **legs**.
  - To find the area, **multiply the legs**, then **divide by 2**.
  - ![The area question of a right-angled triangle with a base of 5 cm and height of 12 cm. Calculations demonstrate the area formula, yielding 30 cm².](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/area-of-a-triangle/3%29%20Right-angled%20triangles%20Example.webp)
  ### 🛎️ Finding the Height from the Area
  - Use the **same formula**: Area = 1/2 × base × height
  - Substitute the known values and **solve for the missing height**.
  - ![How to calculate the height of a triangle using its area, with given dimensions and a worked-out formula solution.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/area-of-a-triangle/4%29%20Finding%20the%20height%20from%20the%20area.webp)
- [Perimeter of a Polygon](https://maths-angel.com/lessons/perimeter-of-a-polygon)
  > The perimeter of a polygon is the total length of all its sides. Learn formulas for rectangles, squares, triangles, and compound shapes. Watch free!
  ### 🛎️ What is Perimeter?
  - The perimeter of a polygon is the **total length of its boundary.**
  - For a **rectangle**: perimeter = 2 × (length + width).
  - For a **square**: perimeter = 4 × side length.
  - ![Perimeter of a rectangle and square with formulas. Rectangle: length 5 cm, width 3 cm, perimeter 16 cm. Square: side 2 m, perimeter 8 m.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/perimeter-of-a-polygon/1%29%20Perimeter%20of%20Polygons%2C%20Quadrilaterals%2C%20Squares.webp)
  ### 🛎️ Perimeter of Triangles
  - The perimeter of a triangle is the **sum of its three sides.**
  - For an **isosceles** triangle: Perimeter = 2 × equal side + base
  - For an **equilateral** triangle: Perimeter = 3 × side length
  - ![Comparing the perimeter of an isosceles triangle and an equilateral triangle, showing the formulas and values for calculating their perimeters.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/perimeter-of-a-polygon/2%29%20Perimeter%20of%20Triangles.webp)
  ### 🛎️ Perimeter of Complex Shapes
  - Add the lengths of the **edges around the outside of the shape.**
  - You can also **add lengths together** to think of the shape **as a rectangle**.
  - ![Perimeter of an L-shaped polygon shown with two calculation methods. Method 1 adds all side lengths. Method 2 treats it as a rectangle.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/perimeter-of-a-polygon/3%29%20Perimeter%20of%20Complex%20Shapes.webp)
- [Area of a Trapezium](https://maths-angel.com/lessons/area-of-a-trapezium)
  > The area of a trapezium is found using (a + b) ÷ 2 × height. Learn with step-by-step examples for right-angled and isosceles trapeziums. Watch free!
  ### 🛎️ What is a Trapezium?
  - A trapezium is a **four-sided shape** with **only one pair of parallel sides**.
  - The **height** is the **perpendicular distance** between the parallel sides.
  - ![Trapezium definition diagram showing bases, height, and difference from a parallelogram.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/area-of-a-trapezium/1%29%20What%20is%20a%20Trapezium.webp)
  ### 🛎️ Area of a Trapezium
  - The area of a trapezium is given by: Area = (a + b)/2 × h
  - Here, a and b are the lengths of the **parallel sides**.
  - ![Diagram showing the trapezium area formula: (a + b) ÷ 2) x h, and example calculation using bases 8 cm and 12 cm, height 6 cm.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/area-of-a-trapezium/2%29%20Area%20of%20a%20Trapezium.webp)
  ### 🛎️ Area of a Right-Angled Trapezium
  - A right-angled trapezium has **two right angles**, so the height is easy to identify.
  - Use the same formula and take the **perpendicular side as the height**.
  - ![Area of a right-angled trapezium formula, with example of a right-angled trapezium with bases 6 cm and 9 cm, height 4 cm, showing result as 30 cm².](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/area-of-a-trapezium/3%29%20Area%20of%20a%20Right-Angled%20Trapezium.webp)
  ### 🛎️ Finding the Height of a Trapezium
  - Use the same formula: Area = (a + b)/2 × h
  - Substitute the known values and **solve for the height**.
  - ![Area calculation of an isosceles trapezium with bases 5 m and 9 m, total area 21 m², showing height calculated as 3 m.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/area-of-a-trapezium/4%29%20Area%20of%20an%20Isosceles%20Trapezium.webp)
- [Area of Parallelograms and Triangles](https://maths-angel.com/lessons/area-parallelograms-triangles)
  > The area of a parallelogram is base × height. A triangle's area is ½ × base × height. Learn to identify the perpendicular height with examples. Watch free!
  ### 🛎️ Height of a Parallelogram
  - The **height** h is **perpendicular** to the **base** b.
  - The height goes from the **base** to the **opposite side**, even if it is outside.
  - ![Height of a parallelogram illustrated with a perpendicular height h from base b to the opposite side.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/area-parallelograms-triangles/1%29%20Height%20of%20a%20Parallelogram.webp)
  ### 🛎️ Area of a Parallelogram
  - The **area** of a parallelogram is found using **base × height**.
  - This is written as Area = b × h
  - ![Diagram showing the formula for the area of a parallelogram, A = b⋅h, with labelled base (b) and height (h) on two parallelograms.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/area-parallelograms-triangles/2%29%20Area%20of%20a%20Parallelogram.webp)
  ### 🛎️ Height of a Triangle
  - The **height** h is **perpendicular** to the **base** b.
  - The height goes from the **base** to the **opposite vertex**.
  - ![Height of a triangle, with height h perpendicular to base b and extending from b to the opposite vertex, illustrated with two triangle diagrams.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/area-parallelograms-triangles/3%29%20Height%20of%20a%20Triangle.webp)
  ### 🛎️ Area of a Triangle
  - The **area** of a triangle is **half** the area of a parallelogram.
  - The formula is A = \dfrac{1}{2} · b · h.
  - ![Formula for the area of a triangle with base and height labelled, showing two triangles with base (b) and height (h).](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/area-parallelograms-triangles/4%29%20Area%20of%20a%20Triangle.webp)
- [Area of Compound Shapes](https://maths-angel.com/lessons/area-of-compound-shapes)
  > Compound shapes combine simpler shapes. Learn to find their area by splitting into rectangles and triangles, then adding or subtracting areas. Watch free!
  ### 🛎️ Area of a Compound Shape
  - A **compound shape** is made by joining **two or more simple shapes**.
  - To find the area, **break it into familiar shapes** such as rectangles, triangles, or circles.
  - ![Diagram explaining how to find the area of a compound shape by breaking it into simpler shapes, then summing or subtracting their areas.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/area-of-compound-shapes/1%29%20How%20to%20Find%20Area%20of%20Compound%20Shape.webp)
  ### 🛎️ - Example: Two Rectangles
  - An **L-shaped figure** can be split into **two rectangles**.
  - Find each rectangle’s area using **length × width**.
  - **Add** the two areas to get the **total area**.
  - ![Diagram showing the area of a compound shape made of two rectangles with dimensions and steps for calculating total area, totalling 12 square metres.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/area-of-compound-shapes/2%29%20Compound%20Shape%20Two%20Rectangles.webp)
  ### 🛎️ Example: Mixed Shapes
  - Find the **area of each shape** separately, such as a **rectangle** and a **triangle**.
  - The **triangle area formula** is A=\tfrac{1}{2}× base× height.
  - **Add** the areas together to get the **total area**.
  - ![Diagram showing a compound shape composed of a rectangle and a right-angled triangle, with their respective and total areas calculation.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/area-of-compound-shapes/3%29%20Compound%20Shape%20Rectangle%20and%20Triangle.webp)
  ### 🛎️ Real-Life Application
  - Always calculate the **total area** before finding **costs**.
  - **Subtract areas** for **holes** like doors or windows.
  - Multiply the area by the **price per square metre**.
  - ![Diagram showing the calculation of the shaded area of a wall, excluding a door and window, with a total painting cost of £46 at £5 per square metre.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/area-of-compound-shapes/4%29%20Application%20of%20Area%20of%20Compound%20Shapes.webp)
- [Circumference and Area of a Circle and a Sector](https://maths-angel.com/lessons/circumference-area-circle-sector)
  > The circumference of a circle is C = 2πr and the area is A = πr². Learn to calculate arc length and sector area with step-by-step examples. Watch free!
  ### 🛎️ π and Circumference
  - π is a number used in circle formulas and is usually rounded to 3.14 in calculations.
  - The formula for circumference is **C = πd** or **C = 2πr**.
  - ![Diagram showing relationship between pi and circumference: a rolling wheel illustrates that circumference equals pi times diameter.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/circumference-area-circle-sector/1%29%20Constant%20pi.webp)
  ### 🛎️ Finding the **Circumference** of a Circle
  - If the **radius** is 5 cm, substitute into **C = 2πr**.
  - This gives **C = 10π cm**, which is approximately **31.4 cm**.
  - ![Circle with radius 5 cm showing formula for circumference C = 2πr. Example calculation gives circumference ≈ 31.4 cm.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/circumference-area-circle-sector/2%29%20Circumference%20of%20a%20Circle.webp)
  ### 🛎️ Finding the **Area of a Circle**
  - The formula for circle area is **A = πr²**.
  - If the **radius** is 3 cm, the area is **9π cm²**, about **28.26 cm²**.
  - ![Area of a circle formula Area equals pi r squared., with an example that the radius is 3 cm. The area is calculated as approximately 28.26 cm².](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/circumference-area-circle-sector/3%29%20Area%20of%20a%20Circle.webp)
  ### 🛎️ **Circle Sectors**
  - A sector is a **fraction of a circle** based on the **central angle**.
  - Use **α/360 × 2πr** for **arc length** and **α/360 × πr²** for **area**.
  - ![Circle sector diagram with formulas: Arc length equals alpha over 360 times 2 r; area equals alpha over 360 times pi r squared.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/circumference-area-circle-sector/4%29%20Arc%20Length%20and%20Area%20of%20Circle%20Sectors.webp)
- [Solids and Units of Volume](https://maths-angel.com/lessons/solids-units-of-volume)
  > Solids are 3D shapes with length, width, and height. Learn common units of volume — mm³, cm³, mL, and L — and how to convert between them. Watch free!
  ### 🛎️ What are Solids?
  - Solids are **3D shapes** with **length, width, and height.**
  - Solids take up **space.** Cubes and cuboids are common solids.
  - ![Rectangular prism with labelled dimensions for length, width, and height, representing a 3D solid shape in geometry.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/solids-units-of-volume/1%29%20Solids.webp)
  ### 🛎️  Parts of Cubes and Cuboids
  - **Faces** are flat surfaces, **edges** are where faces meet, and **vertices** are corners.
  - **A cuboid** has 6 faces, 12 edges, and 8 vertices.
  - ![Showing the properties of solids, focusing on cuboids and cubes, with details on vertices, edges, and faces, comparing rectangle and square faces.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/solids-units-of-volume/2%29%20Cuboid%20and%20Cubes.webp)
  ### 🛎️ What Does Volume Mean?
  - Volume is the **amount of space** a solid occupies.
  - If a solid is made from **27 cubes** of size **1 cm³**, its volume is **27 cm³.**
  - ![Diagram showing volume of a cube: 27 unit cubes (1 cm³ each) form a 3×3×3 cube with total volume of 27 cm³.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/solids-units-of-volume/3%29%20Volume.webp)
  ### 🛎️ Units of Volume
  - Volume is measured using **cubic units** such as **mm³, cm³, and m³.**
  - 1 cm³ is the volume of a cube **1 cm × 1 cm × 1 cm,** about the size of a dice.
  - ![Volume units chart showing cubic millimetre, cubic centimetre, litre, and cubic metre with examples and corresponding edge lengths.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/solids-units-of-volume/4%29%20Units%20of%20Volume.webp)
  ### 🛎️ Converting Between Units of Volume
  - When converting to a **smaller unit, multiply by 1000** (e.g. 5 cm³ = 5000 mm³).
  - When converting to a **larger unit, divide by 1000** (e.g. 2000 ml = 2 L).
  - ![Converting units of volume between cubic millimetres, cubic centimetres, millilitres, litres, and cubic metres with examples.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/solids-units-of-volume/5%29%20Converting%20between%20Units%20of%20Volume.webp)
- [Volume of a Cuboid and Cube](https://maths-angel.com/lessons/volume-of-a-cuboid)
  > The volume of a cuboid is length × width × height, and a cube is side³. Learn to find volumes of cubes, cuboids, and composite 3D shapes. Watch free!
  ### 🛎️ Volume of Cuboids and Cubes
  - The volume of a **cuboid:** Volume = length × width × height.
  - The volume of a **cube:** Volume = side × side × side.
  - ![Diagram showing the volume formulas of a cuboid and a cube. Cuboid: 5 cm x 3 cm x 2 cm, volume 30 cm³. Cube: 5 cm sides, volume 125 cm³.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/volume-of-a-cuboid/1%29%20Volume%20of%20Cuboids%20an%20Cubes.webp)
  ### 🛎️ Finding a Missing Dimension
  - If the **volume of a cuboid** and two **dimensions** are known, the third can be found.
  - Use Volume = length × width × height, then **divide to find the missing value.**
  - ![Solving for width of a cuboid using volume equals length times width times height formula, with given volume 120 m³, length 10 m, height 2 m.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/volume-of-a-cuboid/2%29%20Volume%20of%20Cuboids%20Calculation.webp)
  ### 🛎️ Volume of Complex Solids
  -	A complex solid can be split into **smaller cuboids.**
  -	Find the volume of **each part** and **add them together.**
  - ![Diagram shows the volume calculation of a complex shape divided into two cuboids A and B, with their respective dimensions and total volume summation.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/volume-of-a-cuboid/3%29%20Volume%20of%20Complex%20Solids.webp)
  ### 🛎️ Using Volume in Real Life
  -	Volume can be used to find **capacity,** such as how much liquid a container holds.
  -	You may need to **convert units** (e.g. from cm³ to litres) when calculating volume.
  - ![The application of cuboid volume formula to calculate water container problem, with dimensions 50 cm x 30 cm x 20 cm, using a flow rate of 10 L/min.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/volume-of-a-cuboid/4%29%20Volume%20of%20Cuboid%20Application.webp)
- [Surface Area of Solids](https://maths-angel.com/lessons/surface-area)
  > Surface area is the total area of all faces of a 3D shape. Learn formulas for cubes, cuboids, and prisms with step-by-step worked examples. Watch free!
  ### 🛎️ What is Surface Area?
  - Surface area is the **total area of all the faces** of a solid.
  - To find it, calculate the area of **each face** and **add them together.**
  - ![Surface area calculation of a cuboid with labelled dimensions: length (l), width (w), and height (h). Illustration highlights 6 faces in 3 pairs.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/surface-area/1%29%20Surface%20Area%20of%20Solids.webp)
  ### 🛎️ Surface Area of Cubes and Cuboids
  - **A cuboid** has **6 faces,** arranged in **3 identical pairs.**
  - **A cube** has **6 identical square faces.**
  - ![Illustration showing surface area formulas for a cuboid and a cube. The cuboid formula is 2lw + 2lh + 2hw, and the cube formula is 6a².](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/surface-area/2%29%20Surface%20Area%20of%20Cuboids%20and%20Cubes.webp)
  ### 🛎️ Calculating the Surface Area of a Cuboid
  -	A cuboid has **3 pairs of identical faces:** top and bottom, front and back, and left and right.
  -	Find the area of one face in each pair, **multiply each by 2,** then add them together.
  - ![Diagram illustrating how to find the surface area of a cuboid with dimensions, by breaking down the calculations for each face and sum up.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/surface-area/3%29%20Calculating%20Surface%20Area%20of%20a%20Cuboid.webp)
  ### 🛎️ Calculating the Surface Area of a Cuboid
  -	A prism has **two identical ends** and several **rectangular side faces.**
  -	Find the area of every face, then **add them all together** to get the surface area.
  - ![A trapezoidal prism with labelled dimensions used to calculate its surface area. The faces include front and back triangles, base, top, and side.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/surface-area/4%29%20Calculating%20the%20surface%20area%20of%20a%20Prism.webp)
- [Volume of Prisms and Cylinders](https://maths-angel.com/lessons/volume-prisms-cylinders)
  > The volume of a prism or cylinder is base area × height. Learn to calculate volumes for different cross-sections with step-by-step examples. Watch free!
  ### 🛎️ Volume of Prisms and Cylinders
  - The volume of any prism or cylinder is V = B × h
  - B is the **area of the base**, and h is the **perpendicular height**
  - ![Volume formulas for prisms and cylinders showing the equation V = B × h, where B is the base area and h is the height.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/volume-prisms-cylinders/1%29%20Volume%20of%20Prisms%20and%20Cylinders.webp)
  ### 🛎️ Finding the Volume of a Prism
  - First, find the **area of the base** (e.g., area of the triangle)
  - Then multiply this area by the **height of the prism** (i.e., the distance between the two parallel faces)
  - ![Calculating the volume of a triangular prism using the formula V = B × h, with base dimensions 4 cm by 3 cm, and height 5 cm, resulting in 30 cm³.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/volume-prisms-cylinders/2%29%20Volume%20of%20Prisms.webp)
  ### 🛎️ Finding the Volume of a Cylinder
  - The base of a cylinder is a **circle**, so B = π r²
  - Multiply B by the **height of the cylinder** to get the volume: V = π r² h
  - ![Volume calculation of a cylinder with a radius of 2 cm and height of 5 cm, using the formula V = B × h. The final volume is approximately 62.8 cm³.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/volume-prisms-cylinders/3%29%20Volume%20of%20Cylinders.webp)
- [Cavalieri's Principle](https://maths-angel.com/lessons/cavalieris-principle)
  > Cavalieri's Principle states that solids with the same height and cross-sectional area have equal volume. Learn to apply it to compare volumes. Watch free!
  ### 🛎️ Cavalieri’s Principle in Real Life
  - Imagine two stacks made from the **same number** of slices.
  - If each slice has the **same area**, the stacks have the **same volume**.
  - Rearranging the slices does not change the **volume** of the stack.
  - ![Two equal-height stacks of 10 slices, one straight and one slanted, illustrating Cavalieri's Principle.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/cavalieris-principle/1%29%20An%20Example%20of%20Cavalieri%27s%20Principle.webp)
  ### 🛎️ What Is Cavalieri’s Principle?
  - Two solids can look different but still have the same **volume**.
  - They must have the same **height**.
  - They must have the same **cross-sectional area** at **every level**.
  - ![Two identical stacks of bread slices illustrate Cavalieri's principle, showing that even if one stack is tilted, the total volume remains unchanged.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/cavalieris-principle/2%29%20What%20is%20Cavalieri%27s%20Principle.webp)
  ### 🛎️ Use Cavalieri’s Principle to Find Volume
  - At each **height**, a tilted prism has the same **cross-sectional area** as an upright prism with the same **base area**.
  - So by **Cavalieri’s principle** they have the same **volume**, and you can use **V = base area × height**.
  - ![Comparison of a rectangular prism and a cylinder, both 10 cm tall, illustrating Cavalieri's principle with equal volumes.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/cavalieris-principle/3%29%20Calculating%20Volume%20of%20Inclined%20Solids%20Using%20Cavalieri%27s%20Principle.webp)
  ### 🛎️ Checking Equal Volume Using Cavalieri’s Principle
  - Compare the **height** and the **cross-sectional area** at matching **levels**.
  - If they match all the way up, the solids have equal **volume**.
  - ![Cavalieri’s principle diagram comparing two solids with equal height and cross-sectional areas at every level, explaining they have the same volume.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/cavalieris-principle/4%29%20Check%20Whether%20Two%203D%20Shapes%20Have%20Same%20Volume%20With%20Cavalieri%27s%20Principle.webp)
- [Volume and Surface Area of Pyramids, Cones, Spheres](https://maths-angel.com/lessons/volume-surface-area-pyramids-cones-spheres)
  > Pyramids and cones share the formula V = ⅓ × base area × height. Learn volume and surface area for all three shapes with worked examples. Watch free!
  ### 🛎️ What Is a Pyramid?
  - A **pyramid** is a **3D shape** with a **polygon base** and **triangular faces** meeting at a single point.
  - The base can be a **triangle**, **square**, or any other **polygon**.
  - ![Diagram showing volume and surface area formulas for a pyramid. Includes base area, height, and the net of the pyramid. ](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/volume-surface-area-pyramids-cones-spheres/1%29%20What%20is%20a%20Pyramid.webp)
  ### 🛎️ Volume and Surface Area of a Pyramid
  - The **volume** of a pyramid is V=1/3Bh, where B is the **base area** and h is the **vertical height**.
  - The **surface area** is A=B+L, where L is the **total area of all triangular faces**.
  - ![Illustrating the volume and surface area of a square-based pyramid, including calculations for volume and surface area along with the pyramid's net.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/volume-surface-area-pyramids-cones-spheres/2%29%20Volume%20and%20Surface%20Area%20of%20Pyramids.webp)
  ### 🛎️ Example: Pyramid Volume and Surface Area
  - With a **square base** of side 10 cm and height 12 cm, V=1/3(10²)(12)=400 cm³.
  - Adding the base and four identical **triangular faces** gives a **surface area** of 360 cm².
  - ![Square pyramid with base 10 cm, height 12 cm, slant 13 cm; net shown; volume 400 cm³, surface area 360 cm².](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/volume-surface-area-pyramids-cones-spheres/3%29%20Volume%20and%20Surface%20Area%20of%20Pyramids%20Example.webp)
  ### 🛎️ Volume and Surface Area of a Cone
  - The **volume** of a cone is V=1/3π r²h, where r is the **radius** and h is the **vertical height**.
  - The **surface area** is A=π r²+π rs, where s is the **slant height**.
  - ![The volume and surface area of a cone, showing formulas for volume (V = 1/3 × B × h) and surface area (A = B + L), with net representation.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/volume-surface-area-pyramids-cones-spheres/4%29%20Volume%20and%20Surface%20Area%20of%20Cones.webp)
  ### 🛎️ Example: Cone Volume and Surface Area
  - If r=3 cm and h=5 cm, then V=1/3π(3²)(5)=15π cm³.
  - Using s=6 cm gives a **surface area** of A=π(3²)+π(3)(6)=27π cm².
  - ![Cone with radius 3 cm, height 5 cm, slant height 6 cm. Volume is calculated to be 15π cm³ and surface area is 27π cm².](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/volume-surface-area-pyramids-cones-spheres/5%29%20Volume%20and%20Surface%20Area%20of%20Cones%20Example.webp)
  ### 🛎️ Volume and Surface Area of a Sphere
  - The **volume** of a sphere is V=4/3π r³, so the **radius is cubed**.
  - The **surface area** is A=4π r², so the **radius is squared**.
  - **Exam tip:** Volume uses **cubed units** and area uses **squared units**.
  - ![Cross-section of a sphere with a radius of 3 cm, showing formulas for volume and surface area.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/volume-surface-area-pyramids-cones-spheres/6%29%20Volume%20and%20Surface%20Area%20of%20Spheres.webp)
- [Pythagoras' Theorem](https://maths-angel.com/lessons/pythagoras-theorem)
  > Pythagoras' Theorem states a² + b² = c² for right-angled triangles. Learn to find missing sides, calculate distances, and use it in 3D shapes. Watch free!
  ### 🛎️ Pythagoras' Theorem
  - Only works in a **right-angled triangle**
  - The **square of the hypotenuse** equals the **sum of the squares** of the other two sides: a² + b² = c²
  - ![Diagram explaining Pythagoras' theorem using two right-angled triangles, showing the equation a² + b² = c², and an example showing 3² + 4² = 5².](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/pythagoras-theorem/1%29%20Pythagoras%20Theorem.webp)
  ### 🛎️ Finding a Missing Side in a Triangle
  - Identify the **hypotenuse** first (the side opposite the right angle)
  - Substitute the known lengths into a² + b² = c², then solve
  - ![Right-angled triangle with sides 5 cm, 13 cm, and unknown side x. Using Pythagoras' theorem, x is calculated as 12 cm with steps shown.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/pythagoras-theorem/2%29%20Pythagoras%20Theorem%2C%20missing%20side%20triangle.webp)
  ### 🛎️ Finding the Distance Between Two Points
  - Draw a **right-angled triangle** using horizontal and vertical distances
  - Use Pythagoras to find the **diagonal distance**: d² = x² + y²
  - ![Pythagoras' theorem example showing a right-angled triangle with sides 6, 8, and hypotenuse x. Equation 6² + 8² = x² solves for x = 10.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/pythagoras-theorem/3%29%20Pythagoras%20Theorem%2C%20distance%20two%20points.webp)
  ### 🛎️ Using Pythagoras in 3D Shapes
  - Break the problem into **two right-angled triangles**
  - Apply Pythagoras **twice** to find the **longest diagonal**
  - ![3D Pythagoras theorem example in a cuboid: x² = 4² + 2² + 3², giving diagonal length x = √29.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/pythagoras-theorem/4%29%20Pythagoras%20Theorem%2C%20room%20diagonale%20cuboid.webp)
- [Trigonometry: Sine, Cosine, Tangent](https://maths-angel.com/lessons/trigonometry-sine-cosine-tangent)
  > Trigonometry ratios — sin, cos, tan — help find missing sides and angles in right-angled triangles. Learn SOH CAH TOA with worked examples. Watch free!
  ### 🛎️ What Are sin, cos, and tan?
  - These are **ratios** used in **right-angled triangles**
  - Once the angle is given, the ratios are **fixed**
  - ![Definitions, ratios, and visuals of sine, cosine, and tangent shown in a right triangle.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/trigonometry-sine-cosine-tangent/1%29%20Trigonometry%2C%20Sine%2C%20Cosine%2C%20Tangent.webp)
  ### 🛎️ Using sin, cos, and tan
  - **SOH**: \sinθ = opposite/hypotenuse
  - **CAH**: \cosθ = adjacent/hypotenuse
  - **TOA**: \tanθ = opposite/adjacent
  - ![A right triangle with 30° angle, opposite side 4 cm. Using the ratio of sin 30°, solve the hypotenuse, which is 8 cm.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/trigonometry-sine-cosine-tangent/2%29%20Trigonometry%2C%20Sine%20for%20Finding%20Side.webp)
  ### 🛎️ Finding a Missing Side
  - Choose **sin, cos, or tan** based on the sides you know and the side you want
  - Substitute the values and **rearrange** to find the unknown side
  - ![The application of using tangent to find the missing side of a right triangle with a 50-degree angle and a known side of 5 cm.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/trigonometry-sine-cosine-tangent/3%29%20Trigonometry%2C%20Tangent%20for%20Finding%20Side.webp)
  ### 🛎️ Finding a Missing Angle
  - Write the correct trig ratio first (sin, cos, or tan)
  - Use the **inverse function** on the calculator (\sin⁻¹, \cos⁻¹, or \tan⁻¹)
  - ![Finding the angle using cosine in a right triangle with sides 4 cm and 8 cm, showing the calculation cos theta = 1/2 and theta = cos⁻¹(1/2) = 60°.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/trigonometry-sine-cosine-tangent/4%29%20Trigonometry%2C%20Cosine%20for%20Finding%20Angle.webp)
  ### 🛎️ Common Trigonometric Values
  - You should know the exact values for **30°**, **45°**, and **60°** without a calculator
  - Having these **triangles in your head** helps you see the ratios instantly
  - ![Common values of sine, cosine, and tangent for 30°, 60°, and 45° with right triangles and ratios.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/trigonometry-sine-cosine-tangent/5%29%20Trigonometry%2C%20Common%20values%20Sine%2C%20Cosine%2C%20Tangent.webp)
- [Cosine Rule](https://maths-angel.com/lessons/cosine-rule)
  > The cosine rule (a² = b² + c² − 2bc cos A) finds missing sides and angles in non-right-angled triangles. See worked examples and useful tips. Watch free!
  ### 🛎️ The Cosine Rule
  - The **cosine rule** is used to find a **missing side or angle** in any triangle.
  - It is used when the triangle is **not right-angled** (otherwise use **Pythagoras’ theorem**).
  - ![Cosine Rule formula a² = b² +c² -2bc·cos(A) explained with a triangle diagram labelled with sides a, b, c, and angle A.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/cosine-rule/1%29%20What%20is%20the%20Cosine%20Rule.webp)
  ### 🛎️ Example: The Cosine Rule for Finding a Side
  - If you know **two sides and the angle between them**, you can find the **third side**.
  - The rule is **a² = b² + c² - 2bc\cos(A)**, where **a** is the unknown side and **A** is the angle between **b** and **c**.
  - ![Using the cosine rule to find a side in a triangle with sides 3 cm and 5 cm and an angle of 60 degrees between them.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/cosine-rule/2%29%20How%20to%20Apply%20the%20Cosine%20Rule%20to%20Find%20Sides%20in%20Triangles.webp)
  ### 🛎️ Rearranging the Cosine Rule to Find an Angle
  - If you know **all three sides**, you can find a **missing angle**.
  - Rearrange to **\cos(A) = \dfrac{b² + c² - a²}{2bc}**. Once you find **\cos(A)**, you can find **angle A**.
  - ![Cosine rule formulas for finding angles in a triangle, with labelled triangle and angle A highlighted in pink.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/cosine-rule/3%29%20The%20Cosine%20Rule%20Formula.webp)
  ### 🛎️ Example: Finding an Angle Using the Cosine Rule
  - Substitute the three **side lengths** into the rearranged formula.
  - Once you find **\cos(F)**, you can find **angle F**.
  - Use **\cos⁻¹** on your calculator to find the angle in **degrees**.
  - ![Triangle GEF with sides 4 cm, 3 cm, and 2 cm, applying the cosine rule to find angle F as 47 degrees using a calculator.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/cosine-rule/4%29%20Applying%20the%20Cosine%20Rule%20to%20Find%20Angles%20in%20Triangles.webp)
- [Sine Rule](https://maths-angel.com/lessons/sine-rule)
  > The Sine Rule is a/sin A = b/sin B = c/sin C. Use it to find missing sides and angles in non-right-angled triangles with clear worked examples. Watch free!
  ### 🛎️ What Is the Sine Rule?
  - The **Sine Rule** is used to find **missing sides or angles** in a **non-right-angled triangle**.
  - The formula is \dfrac{\sin A}{a} = \dfrac{\sin B}{b} = \dfrac{\sin C}{c}.
  - ![Sine Rule Formulas for finding missing sides or angles.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/sine-rule/1%29%20Sine%20Rule.webp)
  ### 🛎️ Using the Sine Rule to Find a Side
  - You must know **one angle** and its **opposite side** as a matching pair.
  - Use this known pair to calculate the **missing side** opposite another **known angle**.
  - ![Sine Rule applied to a triangle with angles 30°, 70°, and one unknown angle, showing how to find the opposite side of 4 cm using the sine rule.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/sine-rule/2%29%20Applying%20the%20Sine%20Rule%20to%20find%20Missing%20Sides.webp)
  ### 🛎️ Using the Sine Rule to Find an Angle
  - You must know **one side** and its **opposite angle** as a matching pair.
  - Then apply **\sin⁻¹** to find the **angle**.
  - ![Applying the sine rule to find angles with a triangle, with the equation sin(135°)/10 cm = sin(theta)/5 cm.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/geometry-measures/sine-rule/3%29%20Applying%20the%20Sine%20Rule%20to%20find%20Missing%20Angles.webp)

## Probability & Statistics

- [Tally Marks, Bar Charts, Tables](https://maths-angel.com/lessons/tally-marks-bar-chart)
  > Tally marks group data in fives, bar charts compare amounts visually, and tables organise data clearly. Learn how to use each one with examples. Watch free!
  ### 🛎️ What Are Tally Mark Charts?
  - Tally marks are used to count data **as it is collected** (e.g. number of cars passing).
  - Marks are **grouped in fives** to make totals quicker and avoid miscounting.
  - ![Tally mark chart comparing counts for white, blue, and yellow categories with totals displayed at the bottom.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/tally-marks-bar-chart/1%29%20Tally%20Mark%20Chart.webp)
  ### 🛎️ How to Read a Bar Chart?
  - Bar charts are used to **compare amounts between groups** (e.g. red cars vs blue cars).
  - Read the number on the axis to find the **value of each group**.
  - ![Bar chart comparing the number of white, blue, and yellow cars respectively, illustrating the height of each bar representing the number of cars.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/tally-marks-bar-chart/2%29%20Bar%20Charts.webp)
  ### 🛎️ What Is a Table Used For?
  - Tables organise data so numbers can be read **clearly and exactly**.
  - They save space and are useful when listing many values.
  - ![Table summarising the number of cars by colour: 12 white, 19 blue, and 8 yellow.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/tally-marks-bar-chart/3%29%20Tables.webp)
  ### 🛎️ When to Use Each Type of Chart
  - The same data can be shown in different formats.
  - Use tally charts to **collect data**, tables to **organise it**, and bar charts to **compare it**.
  - ![Comparison of tally mark chart, bar chart, and table for counting and presenting data on car colours (white, blue, yellow) with respective counts.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/tally-marks-bar-chart/4%29%20Tally%20Marks%2C%20tables%2C%20and%20Bar%20Charts.webp)
- [Absolute Frequency and Relative Frequency](https://maths-angel.com/lessons/absolute-relative-frequency)
  > Relative frequency is the absolute frequency divided by the total. Learn to calculate both frequencies and interpret data with tables and pie charts. Watch free!
  ### 🛎️ What Is Absolute Frequency?
  - It tells you the **exact number of times** something happens
  - Adding all **absolute frequencies** gives the **total number of data values**
  - ![Table showing absolute frequencies of how Class A students get to school: 5 walk, 12 take the bus, 3 go by car, 5 cycle. Total count is 25.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/absolute-relative-frequency/1%29%20Absolute%20Frequency.webp)
  ### 🛎️ What Is Relative Frequency?
  - It tells you how big each category is, as a **fraction or percentage**
  - Adding all **relative frequencies** gives **1, or 100%**
  - ![Table comparing absolute frequency, relative frequency, and relative frequency percentages for modes of transport. Total sum equals 1 or 100%.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/absolute-relative-frequency/2%29%20Relative%20Frequency.webp)
  ### 🛎️ How to Calculate Absolute and Relative Frequency?
  - **Relative** frequency = **absolute** frequency of the category ÷ total number
  - **Absolute** frequency = **relative** frequency of the category × total number
  - ![Table showing how 40 students in Class B get to school, with absolute and relative frequencies for four modes.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/absolute-relative-frequency/3%29%20Absolute%20and%20Relative%20Frequency%20Relationship.webp)
- [Pie Chart](https://maths-angel.com/lessons/pie-chart)
  > A pie chart shows data as slices of a circle. Learn how to draw, calculate central angles, and read pie charts with step-by-step examples. Watch free!
  ### 🛎️ What Is a Pie Chart?
  - A **pie chart** shows how each **category** is a part of the **whole**.
  - Each **slice** represents a **fraction or percentage** of the total.
  - ![A pie chart divided into four sections, labelled 40%, 25%, 20%, and 15%, representing different category contributions.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/pie-chart/1%29%20Pie%20Chart.webp)
  ### 🛎️ How to Draw a Pie Chart
  - Find each **central angle** using **frequency ÷ total × 360°**.
  - Draw the **circle**, measure each **angle**, and label the **categories**.
  - ![Pie chart of favourite sports of 100 students, football (40%), basketball (25%), swimming (20%), and tennis (15%). Central angle formula is displayed.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/pie-chart/2%29%20How%20to%20Draw%20a%20Pie%20Chart.webp)
  ### 🛎️ Reading a Pie Chart
  - Use **angle ÷ 360° × total** to find the **number** in a category.
  - Larger **angles** mean a **bigger proportion** of the whole.
  - ![A visual guide on how to read a pie chart, with a formula for calculating category counts.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/pie-chart/3%29%20How%20to%20Read%20a%20Pie%20Chart.webp)
- [Line Graphs and Curve Graphs](https://maths-angel.com/lessons/line-graphs-curve-graphs)
  > Line graphs connect points with straight lines, curve graphs use smooth curves. Learn to draw and interpret both types with clear examples. Watch free!
  ### 🛎️ How to Draw a Line Graph?
  - Plot **each pair of values** from a table as a **dot** on the graph.
  - Join **neighbouring dots** with **straight lines**.
  - ![Line graph showing temperature changes from -3°C to 3°C throughout the day, with instructions on reading and creating line graphs.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/line-graphs-curve-graphs/1%29%20Line%20Graph%20from%20a%20Data%20table.webp)
  ### 🛎️ How to Read a Line Graph?
  - Use the *x-axis* to find the **input** (the input is often time).
  - Read across to the line, then down the *y-axis* to find the **output**.
  - ![Line graph illustrating daily sales for a week with a peak on Friday, highlighting sales of 20 on Tuesday and 80 over the weekend.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/line-graphs-curve-graphs/2%29%20Interpreting%20a%20Line%20Graph.webp)
  ### 🛎️ When to Use a Curve Graph?
  - Use a curve when the relationship is **not linear**.
  - Fit the data points with **one smooth curve**.
  - ![A curve graph illustrating the growth of a plant over five weeks and running speed over age.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/line-graphs-curve-graphs/3%29%20When%20Curve%20Graphs%20are%20More%20Appropriate.webp)
  ### 🛎️ Line Graphs vs. Curve Graphs
  - Line graphs use **straight lines** to show changes over time.
  - Curve graphs use **one smooth curve** for more complex relationships.
  - ![Comparison of line graphs and curve graphs, highlighting differences in representation and complexity of data relationships.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/line-graphs-curve-graphs/4%29%20Line%20vs%20Curve%20Graphs.webp)
- [How to Calculate the Mean, Median, Mode and Range](https://maths-angel.com/lessons/mean-median-mode-range)
  > Learn how to calculate the mean, median, mode and range with clear definitions, formulas, and worked examples. Watch free!
  ### 🛎️ How to Calculate the Mean?
  - **Add up** all the values.
  - **Divide by** how many values there are.
  - ![Calculating the mean by adding all values and dividing by the count, with a worked example using five numbers.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/mean-median-mode-range/1%29%20Mean.webp)
  ### 🛎️ How to Find the Median?
  - Put the numbers in order **from smallest to largest**.
  - Find the **middle value** (or average the two middle values).
  - ![Finding the median by ordering values from smallest to largest and identifying the middle value in the data set.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/mean-median-mode-range/2%29%20Median.webp)
  ### 🛎️ How to Find the Mode?
  - The mode is the value that appears **most often**.
  - There can be more than one mode, or no mode.
  - ![Identifying the mode as the most frequently occurring value in a data set, shown with example numbers.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/mean-median-mode-range/3%29%20Mode.webp)
  ### 🛎️ How to Calculate the Range?
  - Range = **largest** value − **smallest** value.
  - A **small range** means the data is more **consistent**.
  - ![Calculating the range by subtracting the smallest value from the largest value in a data set.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/mean-median-mode-range/4%29%20Range.webp)
- [Median, Mean, Mode and Range from a Frequency Table](https://maths-angel.com/lessons/median-mean-mode-frequency-table)
  > Learn how to find the mean, median, mode and range from a frequency table. Understand each calculation with clear methods and step-by-step examples. Watch free!
  ### 🛎️ Mean from a Frequency Table
  - Multiply each value by its **frequency** and add the results.
  - Divide by the **total frequency** to find the **mean**.
  - ![Calculating mean from a frequency table showing number of pets owned, including values and frequency, resulting in a mean of 1.6.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/median-mean-mode-frequency-table/1%29%20Mean%20from%20a%20frequency%20table.webp)
  ### 🛎️ Median from a Frequency Table
  - Add the frequencies to find the **total number of values (n)**.
  - Use **(n + 1) ÷ 2** to find the median **position**.
  - Example: if n = 15, (15 + 1) ÷ 2 = 8, so the **8th value** is the median.
  - ![Frequency table showing pets owned and their frequency, with the median calculated as 1 using the formula (n+1)/2.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/median-mean-mode-frequency-table/2%29%20Median%20from%20a%20frequency%20table.webp)
  ### 🛎️ Range from a Frequency Table
  - The **range** is the **largest value minus the smallest value**.
  - **Exam tip:** do **not** use frequencies, only use the **values**.
  - ![Frequency table showing pets owned with mode of 1 and range of 9. Pets owned range from 0 to 9, with frequencies of 4, 6, 3, 1, and 1 respectively.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/median-mean-mode-frequency-table/3%29%20Mode%20and%20Range%20from%20a%20frequency%20table.webp)
  ### 🛎️ Mode from a Frequency Table
  - The **mode** is the value that appears the **most often**.
  - It is the **value** with the **highest frequency**.
  - ![Calculations for mean, median, mode, and range of pets owned, based on a frequency table showing values 0 to 9.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/median-mean-mode-frequency-table/4%29%20Overview%20Mean%2C%20Median%2C%20Mode%2C%20Range%20from%20Frequency%20Table.webp)
- [How to Draw and Read a Box Plot](https://maths-angel.com/lessons/box-plot)
  > A box plot shows data distribution using minimum, Q1, median, Q3, and maximum. Learn to draw and read box plots and find the interquartile range. Watch free!
  ### 🛎️ What Is a Box Plot?
  - A **box plot** shows the **distribution** of a data set.
  - It highlights the **spread** and **middle** of the data clearly.
  - ![Box plot definition showing whiskers, quartiles, and median line used to display the distribution of a data set](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/box-plot/1%29%20Box%20Plots.webp)
  ### 🛎️ Creating a Box Plot
  - Find the **minimum**, **Q1**, **median (Q2)**, **Q3**, and **maximum** values.
  - Draw a **box** from **Q1 to Q3** with a line at the **median**.
  - ![Box plot showing minimum value, lower quartile (Q1), median (Q2), upper quartile (Q3), and maximum value, with example values for hours of exercise.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/box-plot/2%29%20Creating%20Box%20Plot.webp)
  ### 🛎️ Interpreting a Box Plot
  - The **interquartile range (IQR)** is **Q3 − Q1** and shows the middle **50%**.
  - The **range** is **maximum − minimum** and shows the full **spread**.
  - ![Box plot showing minimum, maximum, Q1, median, Q3, interquartile range (IQR), and range.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/box-plot/3%29%20Interpreting%20Box%20Plot.webp)
- [Averages from a Grouped Frequency Table](https://maths-angel.com/lessons/grouped-frequency-table)
  > A grouped frequency table organises data into class intervals. Learn to find the modal class, median class, and estimate the mean step by step. Watch free!
  ### 🛎️ What Is a Grouped Frequency Table?
  - A **grouped frequency table** organises data into **class intervals**.
  - Each class shows how many values **fall within that interval**.
  - ![Grouped frequency table showing delivery time intervals in minutes and their corresponding frequencies.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/grouped-frequency-table/1%29%20Grouped%20Frequency%20Table.webp)
  ### 🛎️ How to Find the Modal Class?
  - The **modal class** is the class with the **highest frequency**.
  - It shows where the data occurs **most often**.
  - ![Frequency table of delivery times showing modal class 40 < x ≤ 50 with highest frequency of 10.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/grouped-frequency-table/2%29%20Modal%20Class.webp)
  ### 🛎️ How to Find the Median Class?
  - Add the frequencies to find the **total number of values (n)**.
  - Use **(n + 1) ÷ 2** to find the **median position**.
  - If n = 25, (25 + 1) ÷ 2 = 13, so the class containing the **13th value** is the median class.
  - ![Grouped frequency table for delivery times, showing frequencies and positions to identify the class containing the median.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/grouped-frequency-table/3%29%20Median%20from%20a%20Grouped%20Frequency%20Table.webp)
  ### 🛎️ How to Estimate the Mean?
  - Find the **midpoint** of each class by adding the **lower and upper values** and dividing by **2**.
  - Multiply each **midpoint** by its **frequency** and add the results.
  - Divide this sum by the **total frequency** to estimate the **mean**.
  - ![Grouped frequency table showing delivery time intervals and frequencies, with explanations of the modal class, class with the median, estimated mean.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/grouped-frequency-table/4%29%20Mean%20from%20a%20Grouped%20Frequency%20Table.webp)
- [Calculating Probability](https://maths-angel.com/lessons/calculating-probability)
  > Probability measures how likely an event is, from 0 to 1. Learn to calculate probability with formulas and explore experimental probability with examples. Watch free!
  ### 🛎️ What Is Probability?
  - **Probability** tells us how **likely** an event is to happen.
  - It is written from **0 to 1**, or **0% to 100%**.
  - **0%** means **impossible** and **100%** means **certain**.
  - ![Explanation of probability, including the likelihood of events happening and a scale ranging from impossible (0%) to certain (100%).](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/calculating-probability/1%29%20Characteristic%20of%20a%20Probability.webp)
  ### 🛎️ Calculating Probability
  - **Probability** equals **number of desired outcomes ÷ total number of outcomes**.
  - For example, with **3 red balls** out of **10 balls**, the probability of red is **3/10** or **30%**.
  - ![Explaining probability of drawing a red ball from a bag of 10 balls (3 red, 7 blue) with a probability formula and example calculation.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/calculating-probability/2%29%20Calculating%20Probabilities.webp)
  ### 🛎️ Experimental Probability
  - **Experimental probability** is used when the **true probability is unknown**.
  - You **repeat an experiment** to **estimate** how likely an event is.
  - The more **trials** you do, the **closer** it gets to the true probability.
  - ![Drawing 100 times and getting 37 red and 63 blue. Explanation of experimental probability as estimated from outcomes compared to true probability.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/calculating-probability/3%29%20Estimating%20Probabilities.webp)
- [Addition Rule of Probability and Expected Frequency](https://maths-angel.com/lessons/addition-rule-of-probability-expected-frequency)
  > The addition rule of probability states that for mutually exclusive events, P(A or B) = P(A) + P(B). Learn how to add probabilities and calculate expected frequency with examples.
  ### 🛎️ What Are Equiprobable Events?
  - **Equiprobable events** are events with the **same chance** of happening.
  - When rolling a **fair die**, each number has an **equal probability** of **1/6**.
  - ![Explanation of equiprobable events by using a die and showing that each face has an equal probability of 1/6.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/addition-rule-of-probability-expected-frequency/1%29%20Equiprobable%20Events.webp)
  ### 🛎️ Calculating Probability: Single Event
  - **Probability** equals **number of favourable outcomes ÷ total outcomes**.
  - For example, drawing a **blue ball** from **50 balls** with **25 blue** gives **25/50 = 50%**.
  - ![Drawing coloured balls from a bag with 50 balls, showing 50% for blue and 30% for red.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/addition-rule-of-probability-expected-frequency/2%29%20Calculating%20Probability.webp)
  ### 🛎️ Probability of “OR” Events
  - Use **OR** when **either** event can happen.
  - Add the **favourable outcomes** for each event, then divide by the **total outcomes**.
  - ![Probability of drawing a red or blue ball from 50 balls, with 25 blue, 15 red, and 10 yellow, equals 80%.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/addition-rule-of-probability-expected-frequency/3%29%20Calculating%20Probability%20Example.webp)
  ### 🛎️ Addition Rule for Probability
  - **Mutually exclusive events** cannot happen **at the same time**.
  - For example, a ball cannot be **red and blue** at the same time.
  - When events are mutually exclusive, use **P(A or B) = P(A) + P(B)**.
  - ![Explanation of the probability of drawing blue or red balls from a bag with 50 balls using the addition rule for mutually exclusive events.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/addition-rule-of-probability-expected-frequency/4%29%20Addition%20Rule%20for%20Probabilities.webp)
  ### 🛎️ Expected Absolute Frequency
  - **Expected absolute frequency** is how often an event is **expected** to happen.
  - It equals **probability × number of trials**.
  - It is an **estimate**, so what actually happens can **vary**.
  - ![Calculating the expected absolute frequency using a die, illustrating that the expected absolute frequency of rolling a '3' over 60 trials is 10.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/addition-rule-of-probability-expected-frequency/5%29%20Expected%20Absolute%20Frequency.webp)
- [Probability Tree Diagrams](https://maths-angel.com/lessons/probability-tree-diagrams)
  > Probability tree diagrams show all outcomes step by step. Learn how to draw them, multiply along branches, and solve combined event problems. Watch free!
  ### 🛎️ What Is a Probability Tree Diagram?
  - A **probability tree diagram** shows **all possible outcomes** step by step.
  - It is used for **two-stage events**, like drawing **one ball at a time**.
  - ![Red and blue balls with numbers 3 and 7 beside a bag of 10 balls, illustrating possible outcomes in a probability experiment.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/probability-tree-diagrams/1%29%20What%20is%20a%20Probability%20Trees%20Diagrams.webp)
  ### 🛎️ Drawing the Probability Tree Correctly
  - Write **fractions on every branch** so probabilities are clear.
  - After the first outcome, **update the probabilities** for what is left.
  - ![Probability tree diagram showing two-ball draws without replacement from a bag with 3 red and 6 blue balls.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/probability-tree-diagrams/2%29%20How%20to%20Draw%20Probability%20Trees%20Diagrams.webp)
  ### 🛎️ Finding the Probability of a Single Outcome
  - You **multiply the probabilities** along one path.
  - This gives the probability of **one specific result**.
  - ![Probability tree diagram illustrating the drawing of 9 balls, 3 red and 6 blue, with dependent probabilities and combined event outcomes.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/probability-tree-diagrams/3%29%20Probability%20Tree%20Diagram%20Outcomes.webp)
  ### 🛎️ Finding the Probability of a Combined Event
  - A **combined event** means more than one path fits the question.
  - You **add the probabilities** of paths like **red-red** and **blue-blue**.
  - ![Probability tree diagram for the calculation of drawing two balls of the same colour using equiprobable events and addition rule for probabilities.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/probability-tree-diagrams/4%29%20Finding%20Probabilities%20with%20Probability%20Tree%20Diagram.webp)
  ### 🛎️ Probability of At Least One Event Happening
  - You can **add all paths** where the event happens.
  - Or do **1 − probability of the opposite** to save time and **check your answer**.
  - ![Probability tree diagram showing the calculation of drawing 2 balls of the same colour with red and blue ball probabilities, expected values.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/probability-tree-diagrams/5%29%20Calculating%20Probabilities%20using%20Probability%20Tree%20Diagrams.webp)
- [Sets and Venn Diagrams](https://maths-angel.com/lessons/sets-and-venn-diagrams)
  > A Venn diagram shows relationships between sets. Learn set notation, union and intersection, and how to interpret Venn diagrams with examples. Watch free!
  ### 🛎️ What Is a Set?
  - A **set** is a **collection** of **distinct** objects.
  - The **order** does not matter and there are **no repeated elements**.
  - ![Introduction to sets showing a set A = {1, 3, 5, 7} defined as a collection of distinct objects, with no repeated elements and order doesn't matter.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/sets-and-venn-diagrams/1%29%20Sets%20in%20Maths.webp)
  ### 🛎️ Understanding Venn Diagrams
  - A **Venn diagram** is a **visual representation** of sets and their **relationships**.
  - The **universal set** shows **everything** being considered.
  - ![Venn diagram showing ice cream preferences of 30 students, with 15 liking chocolate, 10 liking vanilla, and 5 not liking either flavour.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/sets-and-venn-diagrams/2%29%20Venn%20Diagram%20(Notation%29.webp)
  ### 🛎️ Intersection and Union
  - The **intersection (∩)** means **AND**, so it is what is in **set A and set B at the same time**.
  - The **union (∪)** means **OR**, so it is what is in **set A or set B or both**.
  - ![Venn diagram showing the intersection and union of students who like chocolate and vanilla flavour, with definitions and notations for set operations.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/sets-and-venn-diagrams/3%29%20Venn%20Diagram%20(Example%29.webp)
  ### 🛎️ Reading Numbers from a Venn Diagram
  - Add **all regions** to find the **total number** in the **universal set**.
  - **Not B** means elements **outside** set B. To find it, **subtract set B** from the **universal set**.
  - ![Venn diagram showing ice cream preferences with 4 liking chocolate, 8 liking vanilla, 10 liking both, and 6 not liking either. Total students are 28.](https://mathangelwebsitestorage.blob.core.windows.net/mathangel-lesson-images/probability-statistics/sets-and-venn-diagrams/4%29%20Venn%20Diagram%20(Practice%29.webp)

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