Math topics for ages 13–14

Number sense first, then the arithmetic and the shapes, measures and data work that school actually examines. Each topic ends in a Practice Zone that builds a fresh question set every visit.

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

21 topics · 126 hands-on stations · 1 free to try

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The Number Families

Every number you have ever written belongs to a family, and the families nest inside one another like boxes. The interesting question is not what is in each box — it is what the box cannot hold.

Naturals are 1, 2, 3, … Integers add zero and the negatives: …−2, −1, 0, 1, 2… Integers are closed under + − ×, but not under ÷: 3 ÷ 2 escapes the box.
A number you can write as p/q where p and q are integers and q ≠ 0. It has nothing to do with being sensible — the word comes from ratio.
A rational number's decimal always terminates (0.375) or recurs (0.428571428571…). It can never wander forever without a pattern.
Do the long division of p ÷ q. Every remainder is smaller than q, so after at most q steps a remainder must repeat — and once a remainder repeats, the digits repeat forever. That is a genuine proof, not a hand-wave.

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Free printable worksheet

Ten questions from Rational & Irrational 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 Rational & Irrational worksheet (PDF)See the whole trail →

A few of the Math topics at this age

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Rational & Irrational
Free

The number line is mostly holes — and √2 falls straight through one

👪 The Number Families · 🧺 Rational or Irrational? · 📜 Proving √2 Is Irrational · 🎯 Chasing √2 With Fractions · 🧀 The Classic Traps · 🎯 Practice Zone
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Exponents & Standard Form

Why a⁰ = 1 is forced on us, and how to write the size of an atom

📐 Where the Index Laws Come From · 🔭 Powers of Ten · 📏 Is It Really Standard Form? · ⚠️ Index Errors Everyone Makes · 👕 Same Value, Different Clothes · 🎯 Practice Zone
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Squares, Cubes & Roots

Perfect squares, surd simplification, and why √(a+b) is never √a + √b

🔍 Patterns Worth Knowing · 📈 Square, Cube, Root · ✂️ Simplifying Surds · 💣 The Root Is Not Linear · 🧊 Perfect Square, Perfect Cube, or Neither? · 🎯 Practice Zone
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Algebraic Expressions

Terms, coefficients, like terms — and why 2a + 3b just sits there

🦴 The Anatomy of an Expression · 👯 Which Terms Are Alike? · 🏁 The 2x vs x² Race · 🗣️ From English to Algebra · ➖ Expanding and the Minus Sign · 🎯 Practice Zone
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Linear Equations

The balance model, inverse operations, and equations with no solution at all

⚖️ Why You Do the Same to Both Sides · 🏗 Solving One, Step by Step · 🎚️ The Balance Tips · 🕳️ Where Solving Goes Wrong · 🏗️ Building the Equation · 🎯 Practice Zone