Number and shape patterns

How can you predict the 100th term without drawing 100 pictures?

Spotting rules in number sequences and growing shape patterns, describing them in words and symbols, and using the rule to predict.

5 depths≈ 250 min of reading81 practice questions

The ladder

Choose how deep to go

Every depth is open. Start where it feels interesting — you can always move back.

Discoverabout 35 minWhat comes next? Meeting patternsMeet repeating and growing patterns in beads, rangoli, calendars and the hundred square. Find the unit, find the difference, describe the rule in words, and use jumps to predict terms far ahead.Owner-accepted · not line-by-line reviewed
Understandabout 45 minRules, terms and sequencesLearn the precise language of sequences, the difference method for finding rules, arithmetic and geometric sequences, square, cube, triangular and Fibonacci numbers, digit patterns, and the rules behind growing matchstick and dot patterns.Owner-accepted · not line-by-line reviewed
Investigateabout 50 minPattern detectives: predict, test, explainInvestigate growing patterns like a detective: predict first, collect small cases, find the rule, test it and explain why it works. Includes far predictions, working backwards, odd sums, Gauss’s pairing, grid tricks and always-sometimes-never reasoning.Owner-accepted · not line-by-line reviewed
Go deeperabout 60 minWhy patterns work: rules, algebra and proofTurn rules into algebra and prove them: why the step becomes the coefficient of n, why odd numbers make squares, sums of powers and cubes, the Indian discovery of the Fibonacci numbers and Meru Prastara, why digit patterns stop, and why patterns that look certain can break.Owner-accepted · not line-by-line reviewed
Extendabout 60 minPattern hunters: puzzles, projects and open questionsTake patterns into the wider world: Lo Shu, Khajuraho and Ramanujan magic squares, tessellations and symmetry, figurate numbers, cycles of last digits and weekdays, the chessboard legend and binary, olympiad problems, patterns in music and careers, projects, and open questions like Collatz.Owner-accepted · not line-by-line reviewed

Practise and apply

Question bank · 81 questions

9 sections from Foundation to Challenge, each with hints and a worked solution — or play a 10-question round for points and stars.

Foundation 15Core 33Stretch 24Challenge 9

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The web

Connected ideas

Follow a connection when you’re curious — some lead into completely different areas.

Number and shape patterns

Related to
Number system
Place-value charts are full of patterns: each place is ten times the one to its right.
Related to
Properties of numbers
Many number patterns — like the sum of consecutive odd numbers — are properties of numbers in disguise.
Related to
Order of operations
A pattern rule such as 3 × n + 1 is an expression — you need the order of operations to use it.
Contrasts with
Prime and composite numbers
Primes famously refuse to follow a simple pattern, unlike even numbers, squares or multiples.
Uses
HCF and LCM
Two repeating cycles line up again after their LCM — the pattern behind blinking lights and bus timetables.
Related to
Shape and space
Growing shape patterns — matchstick squares, dot triangles — are geometry and number at the same time.

Words to know

71 words are explained in these lessons

All vocabulary →

patternruletermsequencerepeating patternrepeating unitgrowing patternshrinking patterndifferencemultipleeven numberodd number+59 more