Analytical topics for ages 13–14

The reasoning no subject teaches directly: spotting patterns, following a rule, working backwards from a result, and telling a sound argument from a plausible one.

Early secondary. Topics carry the weight and the vocabulary of exam work, and expect a child who can hold several ideas at once.

20 topics · 120 hands-on stations · 1 free to try

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The Difference Ladder

Stop guessing at series. Run the Difference Ladder instead: a fixed order of tests, cheapest first. You climb a rung only when the rung below fails.

Write the gaps underneath the terms. If they are constant, the series is arithmetic and the nth term is a + (n−1)d. Two matching gaps are not proof; check every gap you are given.
Gaps not constant? Take the gaps of the gaps. A constant second difference means a quadratic rule of the form an² + bn + c, and that constant equals 2a. Constant third difference means a cubic.
If the gaps are exploding rather than settling, divide each term by the one before it. A constant ratio means a geometric rule. If the ratio itself climbs 2, 3, 4, 5 you are looking at a factorial-style series.
Test the term against n itself: n², n³, n² ± n, n³ − 1, n! and so on. 7, 26, 63, 124 looks wild until you notice it is n³ − 1 for n = 2, 3, 4, 5.

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Free printable worksheet

Ten questions from Advanced Number Series on one page, with an answer key at the bottom — a PDF you can print for homework, a car journey or a classroom. No account needed and nothing to sign up for.

⬇️ Download the Advanced Number Series worksheet (PDF)See the whole trail →

A few of the Analytical topics at this age

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Advanced Number Series
Free

The difference ladder, the ratio test, and the series that defeat both

🏗 The Difference Ladder · 📐 Watch a Quadratic Give Itself Away · ✖️ When the Ladder Fails · 🧵 Woven and Alternating Series · 🩺 Diagnose Before You Solve · 🎯 Practice Zone
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Alphanumeric and Symbol Series

Three streams running side by side, each with its own private rule

🧬 The Three-Stream Split · 🔡 Pure Letter Series · 🧪 Mixed Terms — Three Rules at Once · 🎞️ Watch Three Streams Advance · 🔗 Match the Series to Its Rule · 🎯 Practice Zone
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Complex Analogies

Building a bridge sentence precise enough that only one answer can cross it

🌉 The Bridge Sentence · 🗣️ Verbal Analogies Under Pressure · 🔢 Number and Letter Analogies · 🗂️ Sort by Relation Family · 🔬 Loose Bridge or Tight Bridge? · 🎯 Practice Zone
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Coding-Decoding — Advanced

Cracking an unknown substitution systematically instead of hoping for inspiration

📋 The Alignment Table · ➡️ Step a Cipher · 🕵️ Crack an Unknown Substitution · 🧬 Which Family of Code? · 🏗 The Cracking Procedure · 🎯 Practice Zone
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Blood Relations and Family Trees

A ledger of generations that turns a tangled sentence into a two-second diagram

📒 The Generation Ledger · 🏗 The Ledger Procedure · ↩️ Walk the Sentence Backwards · ➕ Coded Relations · 🏷️ Name the Relation Exactly · 🎯 Practice Zone