This complete printable course begins with the Fibonacci sequence and gradually connects number patterns to geometry, ratios, infinite decimals, continued fractions and numerical curiosities.

The complete course includes:

• Lesson 1: The Fibonacci sequence;
• Lesson 2: Fibonacci rectangles and spirals;
• Lesson 3: The golden ratio;
• Lesson 4: Numbers that go on forever;
• Lesson 5: Infinite fractions;
• Lesson 6: Other magical numbers;
• six standalone teaching handouts with worked examples and grown-up guidance;
• answers and discussion for every lesson;
• the complete Magical Numbers Exercises booklet with practice, reasoning, investigations and challenges.

The resources are designed for flexible use: follow the website alongside the printables, use the handouts independently, or print individual lessons as needed.

Suitable for learners aged 10 and over. Some later algebraic sections can be treated as optional extension material.

☕ Free to download

This resource is completely free. If you find it useful and would like to support the creation of more free mathematics resources, you can support Sumthing Else on Ko-fi.

This is the complete six-part Beyond Curriculum: Magical Numbers course from Sumthing Else.

The exercises are designed to reinforce the ideas from the lesson handouts while also giving learners opportunities to reason, investigate and go further.

The booklet includes practice on:

• the Fibonacci sequence;
• Fibonacci rectangles and spirals;
• the golden ratio;
• terminating, repeating and irrational decimals;
• continued fractions and approximations;
• other magical-number investigations.

Each lesson section includes a mixture of warm-up questions, core practice, reasoning, investigation and challenge tasks. Answers and discussion are collected at the end, with clickable navigation in the digital PDF.

Suitable for learners aged 10 and over.

Use this booklet alongside the Magical Numbers course or as additional independent practice.

☕ Free to download

This resource is completely free. If you find it useful and would like to support the creation of more free mathematics resources, you can support Sumthing Else on Ko-fi.

This lesson explores:

• Kaprekar’s four-digit routine and the constant 6174;
• the 1089 reverse–subtract–reverse–add trick;
• using place value to explain the hidden structure behind 1089;
• perfect numbers and the sum of their proper divisors;
• the difference between observing a pattern and explaining it;
• how mathematical curiosity can lead from a surprising trick to a proof or general rule.

The printable handout contains worked examples, investigations, explanations, grown-up guidance and answers/discussion, and is designed to work without the website.

Suitable for learners aged 10 and over.

This is Lesson 6 and the conclusion of the Beyond Curriculum: Magical Numbers course.

☕ Free to download

This resource is completely free. If you find it useful and would like to support the creation of more free mathematics resources, you can support Sumthing Else on Ko-fi.

Learners meet continued fractions through accessible recursive examples and see how finite truncations give increasingly accurate rational approximations.

This lesson explores:

• reading and building nested fractions;
• continued fractions for the golden ratio;
• successive Fibonacci-ratio approximations;
• continued-fraction approximations to √2;
• comparing the accuracy of different approximations;
• solving a repeating continued fraction using algebra.

The printable handout includes worked examples, learner tasks, visual structure, grown-up guidance and answers/discussion. It is designed to stand alone as well as complement the website lesson.

Suitable for learners aged 10 and over; the algebraic challenge is optional extension material.

This is Lesson 5 of the six-part Beyond Curriculum: Magical Numbers course.

☕ Free to download

This resource is completely free. If you find it useful and would like to support the creation of more free mathematics resources, you can support Sumthing Else on Ko-fi.

Learners compare different kinds of infinite decimal and investigate the difference between repetition and genuine non-repetition.

This lesson includes:

• terminating decimals;
• repeating decimals and recurring blocks;
• rational numbers written as infinite decimals;
• examples such as 1/7 and its repeating pattern;
• irrational decimals such as √2 and π;
• careful reasoning about what “goes on forever” actually tells us.

The printable handout provides explanations, worked examples, investigations, parent/tutor guidance and answers/discussion, so it can be used independently of the online lesson.

Suitable for learners aged 10 and over.

This is Lesson 4 of the six-part Beyond Curriculum: Magical Numbers course.

☕ Free to download

This resource is completely free. If you find it useful and would like to support the creation of more free mathematics resources, you can support Sumthing Else on Ko-fi.

This lesson introduces the golden ratio through geometry and proportion before moving carefully into its algebraic properties.

Learners explore:

• what it means for rectangles to have the same proportions;
• comparing long-side : short-side ratios;
• the defining self-similarity of a golden rectangle;
• the golden ratio φ ≈ 1.618;
• the identity φ² = φ + 1;
• using that identity to simplify higher powers of φ.

The standalone printable handout contains diagrams, worked examples, learner prompts, explanations for younger learners, grown-up guidance and answers/discussion.

Suitable for learners aged 10 and over. The final algebraic section is a natural extension for learners ready for it.

This is Lesson 3 of the six-part Beyond Curriculum: Magical Numbers course.

☕ Free to download

This resource is completely free. If you find it useful and would like to support the creation of more free mathematics resources, you can support Sumthing Else on Ko-fi.

Learners build rectangles from Fibonacci-sized squares, draw the familiar Fibonacci spiral, and compare the ratios of neighbouring terms as they begin to approach a stable value.

This lesson explores:

• constructing Fibonacci rectangles from successive squares;
• drawing quarter-circle arcs to form a Fibonacci spiral;
• connecting side lengths with consecutive Fibonacci numbers;
• calculating ratios such as 34/21, 55/34 and 89/55;
• noticing how these ratios approach about 1.618.

The printable handout includes visual diagrams, worked examples, learner questions, grown-up tips and answers/discussion, and can be used with or without the website.

Suitable for learners aged 10 and over.

This is Lesson 2 of the six-part Beyond Curriculum: Magical Numbers course.

☕ Free to download

This resource is completely free. If you find it useful and would like to support the creation of more free mathematics resources, you can support Sumthing Else on Ko-fi.

Learners build the Fibonacci sequence slowly from the rule “add the previous two”, then explore what the notation means and why the same structure appears in counting problems.

This lesson develops:

• generating Fibonacci numbers from the recursive rule;
• reading notation such as F₅ without confusing subscripts with powers;
• seeing how each new term depends on the two before it;
• using a staircase-counting problem to uncover the same pattern;
• making and testing mathematical observations.

The printable handout is designed to work independently of the website. It includes visual explanations, worked examples, learner prompts, grown-up guidance, a recap and answers/discussion.

Suitable for learners aged 10 and over.

This is Lesson 1 of the six-part Beyond Curriculum: Magical Numbers course.

☕ Free to download

This resource is completely free. If you find it useful and would like to support the creation of more free mathematics resources, you can support Sumthing Else on Ko-fi.

Counting & Combinatorics begins with a simple question: how can we count every possibility without missing one or counting anything twice? Across six lessons, learners progress from systematic lists and tree diagrams to multiplication, arrangements, combinations, Pascal’s triangle and flexible counting strategies.

The course is designed for exploration rather than memorising formulas. Learners are encouraged to organise possibilities, test ideas, compare methods and explain why their counting works.

The complete course includes

• Lesson 1: How many possibilities? — systematic lists, tables and tree diagrams;
• Lesson 2: Choices in stages — the multiplication principle and multi-stage counting;
• Lesson 3: Arrangements — order, factorials and restricted arrangements;
• Lesson 4: Choosing without ordering — combinations, teams and selections;
• Lesson 5: Pascal’s triangle — combination numbers, symmetry and patterns;
• Lesson 6: Clever counting — complements, cases, symmetry, double counting and other strategies;
• printable handouts;
• supplementary exercises;
• answers and discussion.

Suitable for learners aged 10 and over. No formal prerequisites are required, although the lessons are designed to be followed in order.

☕ Free to download

This resource is completely free. If you find it useful and would like to
support the creation of more free mathematics resources, you can

support Sumthing Else on Ko-fi
.

This download brings together the full 6-part Beyond Curriculum: Counting & Combinatorics course.

In Lesson 6 of Counting & Combinatorics, learners bring together the ideas from the course and explore a collection of powerful counting strategies.

They investigate counting the opposite, splitting a problem into cases, using symmetry, counting the same thing in two ways and replacing a complicated object with a simpler one.

This lesson includes

• complementary counting — counting what is easier instead;
• splitting problems into separate cases;
• using symmetry to avoid repeated work;
• double counting and counting the same set in two ways;
• matching complicated objects with simpler objects;
• problems involving coin tosses, codes, handshakes, diagonals and subsets;
• a rectangle-counting challenge;
• answers and discussion;
• supplementary exercises for extra practice and revision.

Suitable for learners aged 10 and over. This lesson draws on ideas developed throughout Lessons 1–5.

☕ Free to download

This resource is completely free. If you find it useful and would like to
support the creation of more free mathematics resources, you can

support Sumthing Else on Ko-fi
.

This is Lesson 6 of the 6-part Beyond Curriculum: Counting & Combinatorics course.