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IB Class 6 Mathematics★ IB MYP Year 1 · Mathematics · Unit: Number Systems, Patterns and Logic

Number Systems, Patterns and Logic — The Language of Mathematics

Numbers are EVERYWHERE. But who INVENTED them? Why do we count in TENS? What is PRIME, and why does it matter? This unit explores the HISTORY, LOGIC, and PATTERNS of numbers — from ancient civilisations to modern computing. IB MYP Year 1 Mathematics.

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By the end of this chapter you'll be able to…

1Explain why we use base-10; trace the origin of zero and the decimal system to ancient India
2Convert between Hindu-Arabic numerals and Roman numerals
3Explain binary (base-2) as the language of computers
4Define prime numbers; use the Sieve of Eratosthenes to find primes up to 100
5Find HCF using prime factorisation to simplify fractions; find LCM to add fractions with unlike denominators
6Perform operations with negative integers; apply rules for multiplying and dividing negatives
7Identify and extend number sequences: triangular, square, Fibonacci
8Apply BODMAS/PEMDAS to evaluate expressions correctly
9Identify and explain patterns in Pascal's Triangle

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Why this chapter matters

This IB MYP Year 1 Mathematics unit establishes number theory and pattern recognition skills that underpin algebra, statistics, and proof throughout MYP. Prime factorisation (every composite number = unique product of primes) is the foundation of HCF/LCM and fraction work. The Fibonacci sequence and its connection to the Golden Ratio (φ≈1.618) appear in IB Extended Essay and HL Mathematics investigations. BODMAS/PEMDAS as the 'grammar of mathematics' is tested in every subsequent exam. The history of number systems (Babylonian base-60, Hindu-Arabic decimal, binary base-2) connects to IB Global Context 'Personal and Cultural Expression' and is a Criterion D: Applying Mathematics in Real-Life Contexts task. Pascal's Triangle with its hidden patterns (triangular numbers, powers of 2, Fibonacci, binomial coefficients) is a classic Criterion B investigation topic.

Before you start — revise these

A 5-minute refresher here will save you 30 minutes of confusion below.

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IB PYP Year 5: number operations and fractions basics

PYP builds multiplication, division, and basic fractions — MYP Year 1 extends to prime factorisation, HCF/LCM, and negative integers

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IB PYP Year 5: basic patterns and sequences

Visual and number patterns in PYP lead to formal Fibonacci and nth term analysis in MYP

Number Systems, Patterns and Logic

MYP Unit Framework

Key Concept: LOGIC Related Concepts: Patterns. Systems. Representation. Global Context: Personal and Cultural Expression (How do different cultures represent number and quantity?) Statement of Inquiry: Numbers are a UNIVERSAL LANGUAGE, but the WAY we represent and use them reflects the LOGIC, CULTURE, and CREATIVITY of the societies that created them.

Inquiry Questions

Type	Question
Factual	What is a prime number? How does the base-10 system work?
Conceptual	Why do different civilisations develop different number systems? What makes a pattern 'mathematical'?
Debatable	Is mathematics DISCOVERED (always existed, waiting to be found) or INVENTED (created by humans)?

1. How Different Cultures Count — A Brief History

Base-10 (Decimal) — Why 10?

Because we have TEN FINGERS. Counting on fingers = the ORIGINAL calculator. The ancient Hindus developed the decimal system and the concept of ZERO — 'one of the greatest inventions in human history.'

Babylonian Base-60

The Babylonians counted in SIXTIES. That's why we have: 60 seconds in a minute. 60 minutes in an hour. 360 degrees in a circle. 'The Babylonians left their number system embedded in our clocks and compasses.'

Roman Numerals

I=1. V=5. X=10. L=50. C=100. D=500. M=1000. 'Try multiplying CXLII by LXIX in Roman numerals. Now you understand why the Hindu-Arabic system won!'

Binary (Base-2) — The Language of Computers

Only TWO digits: 0 and 1. Every computer on Earth uses binary. '01001000 01101001 = "Hi" in binary. The entire digital world — from your phone to the internet — is built on 0s and 1s.'

2. Number Properties — The Building Blocks

Prime Numbers

A number greater than 1 that has EXACTLY TWO factors: 1 and itself. 2, 3, 5, 7, 11, 13, 17, 19...

'Prime numbers are the ATOMS of mathematics. Every number can be expressed as a UNIQUE product of primes.' Example: 84 = 2² × 3 × 7.

The Sieve of Eratosthenes (c. 240 BCE)

The ancient Greek mathematician found a method to IDENTIFY all primes up to a limit. 'Write all numbers. Cross out multiples of 2. Cross out multiples of 3. Continue. What's left? PRIMES.'

HCF and LCM — Why They Matter

HCF (Highest Common Factor): Used to SIMPLIFY fractions. 12/16 = 3/4 (HCF of 12 & 16 = 4).
LCM (Lowest Common Multiple): Used to find COMMON DENOMINATORS. Add 1/6 + 1/8: LCM of 6 & 8 = 24. Convert: 4/24 + 3/24 = 7/24.

3. Integers — The Positives, the Negatives, and Zero

The Number Line

'Zero is not "nothing." Zero is a NUMBER. It sits at the CENTRE of the number line. Positive numbers to the RIGHT. Negatives to the LEFT.'

Rules for Operations With Negatives

Addition: 5 + (−3) = 2. −5 + 3 = −2. 'Think: movement on the number line.'
Multiplication/Division: (−) × (−) = (+). (−4) × (−3) = +12. 'A negative TIMES a negative = a positive. WHY? Think of it as the OPPOSITE of the opposite.'

Real-World Integers

Temperature (below zero). Elevation (below sea level). Bank balance (overdraft). 'Integers are not abstract. They are how we MEASURE the real world — above and below, credit and debt, profit and loss.'

4. Number Patterns and Sequences

Triangular Numbers: 1, 3, 6, 10, 15...

Tₙ = n(n+1)/2. 'Visualise them as bowling pins.' The 4th triangular number = 4×5/2 = 10.

Square Numbers: 1, 4, 9, 16, 25...

'Draw a square of dots. n × n dots.'

Fibonacci Sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55...

Each term = sum of the TWO previous terms. 'The Fibonacci sequence appears in NATURE: the spiral of a sunflower. The arrangement of pine cones. The breeding pattern of rabbits (the problem Fibonacci was originally solving!). The ratio of consecutive terms approaches the GOLDEN RATIO (φ ≈ 1.618) — called the "divine proportion" by artists and architects because of its aesthetic beauty.'

Pascal's Triangle — Hidden Patterns

'Every number is the SUM of the two numbers above it. The triangle contains: triangular numbers. Powers of 2. Fibonacci numbers (hidden in the diagonals). Binomial coefficients. "Pascal's triangle is a TREASURE MAP of number patterns."'

5. Order of Operations — The Grammar of Mathematics

BODMAS/PEMDAS: Brackets → Orders (Exponents) → Division/Multiplication (left to right) → Addition/Subtraction (left to right). 'The order of operations is the GRAMMAR of mathematics. Without it: 3 + 4 × 5 could be 35 — or 23. The RULE says: multiply first. 3 + 4×5 = 3 + 20 = 23.'

Your Summative Assessment

Task: 'The Number System Investigation'

Choose an ANCIENT CIVILISATION (Babylonian, Mayan, Egyptian, Roman, Chinese).
Research their number system. How did they represent numbers? What BASE did they use?
Compare it with our DECIMAL system. What are the ADVANTAGES of our system? What did the ancient system do BETTER or DIFFERENTLY?
Present your findings. Show calculations in BOTH systems.

ATL Skills

Skill	Focus
Critical Thinking	Analysing patterns. Evaluating different number systems.
Communication	Explaining mathematical reasoning clearly.
Research	Investigating the history and development of number systems.

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

MYP Statement of Inquiry

Numbers are a UNIVERSAL LANGUAGE — but the way we represent them reflects the LOGIC, CULTURE, and CREATIVITY of the societies that created them

Key Concept: LOGIC. Related Concepts: Patterns, Systems, Representation. Global Context: Personal and Cultural Expression.

Prime number definition

Integer > 1 with EXACTLY TWO factors: 1 and itself

First 10 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Note: 1 is NOT prime (only one factor). 2 is the ONLY even prime.

Prime factorisation (Fundamental Theorem of Arithmetic)

Every integer > 1 = UNIQUE product of prime factors (e.g. 84 = 2² × 3 × 7)

Use factor trees. Uniqueness means there is only ONE way to express each number as a product of primes.

HCF from prime factorisation

HCF = product of LOWEST powers of COMMON prime factors

HCF(12, 18): 12=2²×3, 18=2×3². Common primes: 2 (min power: 2¹) and 3 (min power: 3¹). HCF = 2×3 = 6.

LCM from prime factorisation

LCM = product of HIGHEST powers of ALL prime factors

LCM(12, 18): All primes: 2 (max: 2²) and 3 (max: 3²). LCM = 4×9 = 36. CHECK: HCF × LCM = product of original numbers (12×18 = 6×36 = 216 ✓).

Triangular numbers formula

Tₙ = n(n+1)/2

T₁=1, T₂=3, T₃=6, T₄=10, T₅=15. Visualise as triangles of dots (bowling pins).

Square numbers formula

Sₙ = n²

1, 4, 9, 16, 25, 36... Difference between consecutive square numbers = consecutive odd numbers (1, 3, 5, 7...).

Fibonacci sequence rule

F(n) = F(n−1) + F(n−2), with F(1)=1, F(2)=1

Sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55... Consecutive ratio approaches Golden Ratio φ ≈ 1.618.

BODMAS/PEMDAS order

Brackets → Orders/Exponents → Division/Multiplication (left→right) → Addition/Subtraction (left→right)

3 + 4×5 = 3 + 20 = 23. NOT 35. Division and Multiplication are EQUAL priority — do left to right.

Pascal's Triangle row sum

Sum of row n = 2ⁿ

Row 0=1, Row 1=2, Row 2=4, Row 3=8, Row 4=16, Row 5=32. Each element = sum of two directly above it.

Binary to decimal conversion

Each binary digit has positional value: 2⁰=1, 2¹=2, 2²=4, 2³=8, 2⁴=16...

1011 in binary = 1×8 + 0×4 + 1×2 + 1×1 = 11 in decimal.

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Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Treating 1 as a prime number

✓ A PRIME has EXACTLY TWO factors: 1 and itself. The number 1 has only ONE factor (itself). So 1 is NOT prime — it is in its own special category. This preserves the Fundamental Theorem of Arithmetic: if 1 were prime, then 6 = 2×3, but also 6 = 1×2×3 and 6 = 1×1×2×3 — no longer a UNIQUE factorisation. By excluding 1 from primes, every number has exactly ONE prime factorisation.

WATCH OUT

✗ Applying BODMAS in the wrong order — doing addition before multiplication when it appears first left-to-right

✓ BODMAS says: Multiplication and Division come BEFORE Addition and Subtraction, regardless of left-to-right order. 3 + 4×5 = 3 + (4×5) = 3 + 20 = 23. NOT (3+4)×5 = 35. Exception: BRACKETS override everything. (3+4)×5 = 7×5 = 35. Also remember: Division and Multiplication are EQUAL priority — do them left to right. So 12 ÷ 4 × 3 = 3 × 3 = 9 (not 12 ÷ 12 = 1).

WATCH OUT

✗ Confusing HCF (for simplifying fractions) with LCM (for adding fractions)

✓ HCF (Highest COMMON Factor) is the BIGGEST number that divides into BOTH numbers. Use it to SIMPLIFY fractions: 12/18 = 12÷6 / 18÷6 = 2/3. LCM (Lowest COMMON Multiple) is the SMALLEST number that is a multiple of BOTH. Use it to ADD fractions: 1/6 + 1/8 → LCM(6,8)=24 → 4/24 + 3/24 = 7/24. Memory aid: HCF = shrink (simplify). LCM = grow (common denominator).

Practice problems

Try each one yourself before tapping "Show solution". Active recall > rereading.

Q1EASY· prime-factorisation-hcf-lcm

(a) Find the prime factorisation of 360 and 252. Hence find HCF(360, 252) and LCM(360, 252). Verify your answer. (b) Use your LCM to add the fractions 7/360 + 5/252.

▶ Show solution

(a) PRIME FACTORISATION:
360 = 2 × 180 = 2 × 2 × 90 = 2² × 45 = 2² × 9 × 5 = 2³ × 3² × 5.
252 = 2 × 126 = 2 × 2 × 63 = 2² × 63 = 2² × 9 × 7 = 2² × 3² × 7.

HCF = product of LOWEST powers of COMMON prime factors.
Common primes: 2 (min power: 2²) and 3 (min power: 3²).
HCF(360, 252) = 2² × 3² = 4 × 9 = 36.

LCM = product of HIGHEST powers of ALL prime factors.
All primes present: 2 (max: 2³), 3 (max: 3²), 5 (max: 5¹), 7 (max: 7¹).
LCM(360, 252) = 2³ × 3² × 5 × 7 = 8 × 9 × 5 × 7 = 2520.

VERIFY: HCF × LCM = 36 × 2520 = 90,720. Also: 360 × 252 = 90,720. ✓

(b) ADDING FRACTIONS:
7/360 + 5/252. Common denominator = LCM = 2520.
7/360 = 7 × 7 / 2520 = 49/2520. (2520 ÷ 360 = 7)
5/252 = 5 × 10 / 2520 = 50/2520. (2520 ÷ 252 = 10)
49/2520 + 50/2520 = 99/2520.
Simplify: HCF(99, 2520). 99 = 9 × 11. 2520 = 2³ × 3² × 5 × 7. Common factor = 9. 99/9 = 11. 2520/9 = 280.
Final answer: 11/280.

Q2EASY· pascal-triangle-patterns

Write the first 6 rows of Pascal's Triangle (rows 0–5). (a) Find the sum of each row. What pattern do you notice? (b) Where do the triangular numbers appear? (c) Predict the sum of row 10 without writing it out.

▶ Show solution

PASCAL'S TRIANGLE (rows 0–5):
Row 0:          1          → sum = 1
Row 1:         1  1        → sum = 2
Row 2:        1  2  1      → sum = 4
Row 3:       1  3  3  1   → sum = 8
Row 4:      1  4  6  4  1  → sum = 16
Row 5:    1  5  10  10  5  1 → sum = 32

(a) ROW SUM PATTERN:
Sums: 1, 2, 4, 8, 16, 32 = 2⁰, 2¹, 2², 2³, 2⁴, 2⁵ = POWERS OF 2.
Rule: Sum of row n = 2ⁿ.
Why? Each element in row (n−1) contributes to EXACTLY TWO elements in row n (once left-shifted, once right-shifted). So the total DOUBLES each row.

(b) TRIANGULAR NUMBERS:
They appear in the THIRD DIAGONAL (from either side): 1, 3, 6, 10, 15, 21...
In rows 2, 3, 4, 5, 6, 7: the second number from the end of each row (reading the diagonal): 1, 3, 6, 10, 15, 21.
This is the sequence Tₙ = n(n+1)/2.

(c) PREDICTION:
Sum of row 10 = 2¹⁰ = 1024.

Q3MEDIUM· bodmas-sequences

Evaluate: 5 + 3 × (8 − 2)² ÷ 4 − 1. Show every step. Then explain how the Fibonacci sequence connects to the Golden Ratio, and give ONE real-world example where the Golden Ratio appears.

▶ Show solution

BODMAS EVALUATION:
Expression: 5 + 3 × (8 − 2)² ÷ 4 − 1

Step 1 — BRACKETS: (8 − 2) = 6.
→ 5 + 3 × 6² ÷ 4 − 1

Step 2 — ORDERS (exponents): 6² = 36.
→ 5 + 3 × 36 ÷ 4 − 1

Step 3 — MULTIPLICATION AND DIVISION (left to right): 3 × 36 = 108; then 108 ÷ 4 = 27.
→ 5 + 27 − 1

Step 4 — ADDITION AND SUBTRACTION (left to right): 5 + 27 = 32; then 32 − 1 = 31.

FINAL ANSWER: 31.

FIBONACCI AND THE GOLDEN RATIO:
The Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89...

Consider the RATIO of consecutive terms:
2/1 = 2.000
3/2 = 1.500
5/3 = 1.667
8/5 = 1.600
13/8 = 1.625
21/13 = 1.615
34/21 = 1.619
55/34 = 1.618...

As n → ∞, the ratio F(n+1)/F(n) → φ ≈ 1.6180339... (Golden Ratio).

The Golden Ratio φ satisfies: φ² = φ + 1 (i.e. φ = (1+√5)/2).

REAL-WORLD EXAMPLE: Sunflower seed heads — the number of clockwise and anti-clockwise spirals are ALWAYS consecutive Fibonacci numbers (typically 34 and 55, or 55 and 89). This mathematical pattern minimises wasted space and maximises seed packing efficiency — it evolved through natural selection because it is optimal. The Fibonacci/Golden Ratio pattern also appears in pinecones (8 and 13 spirals), pineapples, and the shell of the nautilus.

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•Prime number: exactly TWO factors (1 and itself). 1 is NOT prime. 2 is the ONLY even prime. First 10 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
•Fundamental Theorem of Arithmetic: every integer > 1 has a UNIQUE prime factorisation. Find it using factor trees.
•HCF = lowest powers of COMMON prime factors (use to SIMPLIFY fractions). LCM = highest powers of ALL prime factors (use to ADD fractions).
•Number sequences: Triangular Tₙ = n(n+1)/2 (1, 3, 6, 10...); Square Sₙ = n² (1, 4, 9, 16...); Fibonacci F(n)=F(n-1)+F(n-2) (1,1,2,3,5,8,13...).
•Fibonacci consecutive ratio → Golden Ratio φ ≈ 1.618. Appears in sunflower spirals, pinecones, nautilus shells.
•BODMAS: Brackets → Orders → ÷× (left→right) → +− (left→right). Multiplication is NOT always before division — equal priority, go left to right.
•Pascal's Triangle: each number = sum of two above. Row n sum = 2ⁿ. Third diagonal = triangular numbers. Row sums = powers of 2.
•Binary base-2: only 0 and 1. Positional values: 1, 2, 4, 8, 16... Every computer uses binary.
•Roman numerals: I=1, V=5, X=10, L=50, C=100, D=500, M=1000. SUBTRACTION RULE: smaller before larger = subtract (IV=4, IX=9, XL=40, XC=90, CD=400, CM=900).

IB marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Typical chapter weightage: As per board pattern

Question type	Marks each	Typical count	What it tests
Criterion A (Knowing)	0–8	1 task	Formula application, definitions, calculations
Criterion B (Investigating)	0–8	1 task	Pattern investigation with real data
Criterion C (Communicating)	0–8	1 task	Clearly shown working, appropriate graphs
Criterion D (Applying in Context)	0–8	1 task	Real-world problem with justification

Prep strategy

IB rewards PROCESS over answer — show all working steps with justification
For Criterion D: always link maths to a real context and justify your method choice
Use appropriate mathematical notation throughout
For Criterion B: collect real data (n≥30) and investigate patterns systematically

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

Binary code in every computer and phone

Every piece of digital information — text, images, video, music — is stored and processed as combinations of 0s and 1s (binary). The string '01001000 01101001' encodes 'Hi' in ASCII. Understanding binary is the first step to understanding computing, cryptography, and digital literacy.

Babylonian base-60 in everyday time

Every time you look at a clock you are using Babylonian mathematics: 60 seconds in a minute, 60 minutes in an hour, 360 degrees in a circle. The Babylonians used base-60 because 60 has many factors (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60) — making division easy without fractions.

HCF and LCM in scheduling and engineering

LCM solves real scheduling problems: 'Bus A comes every 6 minutes, Bus B every 8 minutes. When will they both arrive at the same time?' LCM(6,8) = 24 minutes. In engineering, gear ratios and signal synchronisation use the same mathematics.

Golden Ratio in design and art

The Golden Ratio (φ≈1.618) derived from Fibonacci was deliberately used by Renaissance artists (Leonardo da Vinci, in the proportions of the human body in Vitruvian Man) and architects (the Parthenon's facade ratios). Modern designers use it in logo proportions, screen aspect ratios, and typography.

Cryptography and internet security

The security of internet banking depends on the difficulty of factoring very large numbers into primes. RSA encryption (used in HTTPS, the padlock in your browser) uses two enormous primes multiplied together as a public key. Factoring their product is computationally infeasible — this is why prime numbers are foundational to modern cybersecurity.

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

Criterion A (Knowing): For HCF and LCM, ALWAYS show the prime factorisation method — not just the answer. Show the factor tree. Circle the common/all prime factors. Calculate step by step. Examiners give method marks.
Criterion B (Investigating Patterns): For Pascal's Triangle and Fibonacci investigations, go beyond describing what you see. EXPLAIN why the pattern occurs. Predictions must be JUSTIFIED with the rule formula, not just extended by intuition.
Criterion C (Communicating): Show EVERY BODMAS step on a separate line. Do not combine steps or skip to the final answer. Each intermediate step must be visible. For number conversion questions, show the working (e.g. binary positional values table).
Criterion D (Applying in Real Contexts): The 'Number System Investigation' summative task requires you to COMPARE systems (e.g. decimal vs Roman numerals). Structure your comparison: what does each system do well? What are its limitations? Use specific examples (e.g. 'multiplying CXLII by LXIX in Roman numerals is nearly impossible; the same in decimal takes seconds').
Exam time: factor trees and Sieve of Eratosthenes questions take longer than they look. Allocate 4-5 minutes for prime factorisation of two large numbers (for HCF/LCM). Do these first while your mind is fresh.

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

Goldbach's Conjecture (1742): every even integer greater than 2 can be expressed as the sum of two primes. E.g. 28 = 5+23 = 11+17. This conjecture has never been proven or disproven — it remains one of the oldest unsolved problems in mathematics.
Perfect numbers: a number is 'perfect' if it equals the sum of its proper divisors. 6 = 1+2+3. 28 = 1+2+4+7+14. The next perfect number is 496. Are there infinitely many perfect numbers? Unknown. All known perfect numbers are even — are there odd perfect numbers? Also unknown.
Fibonacci in music: the number of notes in musical scales often follows Fibonacci patterns. An octave has 13 notes (8 white, 5 black). In a chord: 3 notes. Composers like Bach and Debussy are claimed to have used Golden Ratio proportions in their compositions.
Pascal's Triangle and probability: row n of Pascal's Triangle gives the number of ways to choose 0, 1, 2... items from n. This connects to IB DP Mathematics' Binomial Distribution and combinatorics (C(n,r) = nCr). Exploring this now gives an enormous advantage in DP Maths HL.

Where else this chapter is tested

CBSE board isn't the only one — other exams test this chapter too.

IB MYP eAssessment — Mathematics (Years 4-5)Direct — prime factorisation, HCF/LCM, BODMAS, and sequences are all assessed in the MYP on-screen mathematics examination

IB DP Mathematics: Analysis and Approaches SL/HLFoundation — number theory (primes, sequences, series), proof by induction, and the Binomial Theorem all extend the concepts introduced here

IB DP Mathematics: Applications and Interpretation SL/HLConnected — statistical distributions and real-life modelling extend the pattern-investigation skills developed in this unit

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Questions students ask

The real ones — pulled from the Q&A community and tutor sessions.

A prime has EXACTLY TWO factors: 1 and itself. The number 1 has only ONE factor — itself. But the deeper reason is the Fundamental Theorem of Arithmetic: every number should have a UNIQUE prime factorisation. If 1 were prime, then 6 = 2×3, but also 6 = 1×2×3, and 6 = 1×1×1×2×3 — no longer unique. Excluding 1 from primes preserves this uniqueness, which is mathematically essential.

HCF (SHRINK): Use to simplify fractions. Find the biggest number that divides both numerator and denominator. 48/72 → HCF(48,72)=24 → 48÷24 / 72÷24 = 2/3. LCM (GROW): Use to add or subtract fractions with different denominators. 1/4 + 1/6 → LCM(4,6)=12 → 3/12 + 2/12 = 5/12. Memory trick: HCF = what's in COMMON (shared factors). LCM = what COVERS both (product if no common factors).

For 6-8 marks, a Criterion B answer must: (1) EXTEND the pattern correctly and show verification, (2) DESCRIBE the pattern in words (not just list numbers), (3) FIND the rule or formula, and (4) JUSTIFY or explain WHY the pattern works. For Pascal's Triangle, showing 'row sums are powers of 2' is good; explaining WHY (each element contributes to exactly two in the next row, so the total doubles) is what earns 7-8 marks.

It is genuinely in nature — and there is a mathematical reason. Plants grow by adding new parts to existing structures in a spiral. The most space-efficient angle for this growth is approximately 137.5° (the 'golden angle', derived from φ). This angle naturally produces Fibonacci numbers of spirals — not by design, but because it is the OPTIMAL packing solution. Natural selection favours plants that pack seeds or leaves most efficiently, so this pattern evolved independently in many species.

Step through in order and BRACKET your work as you go. First, identify ALL brackets and evaluate them. Then, ALL exponents. Then, scan left-to-right for multiplication AND division — evaluate each as you encounter them. Then scan left-to-right for addition AND subtraction. Common trap: '12 ÷ 4 × 3' — do NOT do 4×3=12 first. Do 12÷4=3 first (left to right), then 3×3=9. Another trap: '−3²' = −(3²) = −9, not (−3)² = 9. The negative sign is NOT inside the exponent unless brackets show it.