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🔢 Problem-solving maths for kids

Exponents, roots, fractions, ratios, linear and quadratic equations, sequences, counting, probability, geometry, complex numbers and proof — in the order a problem-solving course teaches them, not the order a textbook lists them.

Start the Maths track

How this track is kept honest

An answer here is a value, not a string: 1/2, 0.5 and 2/4 all count, and so do 2x + 3 and 3 + 2x. Every answer in the course is re-derived independently by sympy before it ships, because a wrong answer in a maths course marks a correct child wrong forever. Figures are drawn to scale, so a child who measures the picture finds it telling the truth.

No programming at all. Roughly ages 9–15, from powers up to proof.

How it starts

Writing 3 × 3 × 3 × 3 is tiring, and easy to get wrong. So we write 3^4 instead. The big number is the base — the thing being multiplied. The little one is the exponent — how many of them there are. So 3^4 means four 3s multiplied together. It does not mean 3 × 4.

The full Maths syllabus

28 units and 200 lessons, in the order they are played. Each unit has a page of its own with every explanation in it, the code samples and what they print, and a few questions to try.

Unit 1: Powers

Exponents, and logarithms backwards

  1. What a Power Means
  2. The Rules Nobody Has to Memorise
  3. Powers of Ten
  4. Roots Are Powers Too
  5. Powers of Whole Brackets
  6. Logarithms Read It Backwards
  7. The Rules of Logarithms

Unit 2: Impossible Numbers

The square root of minus one

  1. Inventing i
  2. Two Numbers in One
  3. The Cycle of Four
  4. Taking Away and Squaring
  5. A Number With an Address
  6. Conjugate Tricks
  7. Roots of Negative Numbers

Unit 3: Finding x

Equations, and keeping the balance

  1. The Balance
  2. Two Steps and Two Sides
  3. Getting Rid of Fractions
  4. Rearranging a Formula
  5. Making One Unknown Vanish
  6. Swapping One Letter In
  7. Turning Words Into x

Unit 4: In Proportion

Ratios, scaling and percentages

  1. Ratios
  2. Growing Together
  3. Three Parts at Once
  4. Rates
  5. Maps and Models
  6. Mixing and Sharing
  7. Percentages Are Ratios Too

Unit 5: Whole Numbers

Factors, primes and remainders

  1. What Goes Into What
  2. Primes: the Building Blocks
  3. More Tests for Dividing
  4. Square Numbers Are Odd
  5. Minus Times Minus
  6. The Last Digit
  7. Sharing and Left Overs

Unit 6: Two Answers

Quadratic equations

  1. Opening and Closing Brackets
  2. Something Times Something Is Zero
  3. A Number in Front
  4. Tidy Up Before You Solve
  5. The Shape of a Quadratic
  6. Stories That Need Two Answers
  7. When It Will Not Factor

Unit 7: Patterns and Tricks

Factorizations worth recognising

  1. The Difference of Two Squares
  2. Squares and Cubes of Brackets
  3. Squares Hiding in Plain Sight
  4. Cubes, Both Ways
  5. Cancelling in Algebra
  6. Substitution Tricks
  7. Looking Before Calculating

Unit 8: Kinds of Number

Where each one came from

  1. Building the Number Line
  2. The Numbers With No Fraction
  3. Working With Fractions
  4. Decimals and Rounding
  5. Surds Behave Like Letters
  6. Naming the Number
  7. The Whole Family

Unit 9: Circles

Round things, and the number pi

  1. All the Way Round
  2. The Space Inside
  3. Working Backwards
  4. The Lines You Can Draw
  5. Wheels and Turning
  6. Circles Meeting Straight Lines
  7. Where Pi Comes From
  8. Slices and Arcs

Unit 10: Angles

Turning, and how much

  1. How Much of a Turn
  2. Crossing Lines
  3. Naming and Measuring
  4. Bearings
  5. Hands on a Clock
  6. Angles With Letters
  7. Angles That Add Up

Unit 11: Triangles

Three sides, and everything they decide

  1. Always One Hundred and Eighty
  2. Pythagoras
  3. Equal Sides, Equal Angles
  4. Triples Worth Knowing
  5. Which Sides Can Make a Triangle
  6. Lines Inside a Triangle
  7. Area From the Three Sides
  8. Same Shape, Same Size

Unit 12: Four Sides

Squares, rectangles and their relatives

  1. The Family Tree
  2. What Each One Promises
  3. All the Way Round
  4. Kites and Trapeziums
  5. Four Sides on a Grid
  6. Proving What a Shape Is
  7. How Much Space Inside

Unit 13: Many Sides

Polygons and their angles

  1. Naming the Shapes
  2. Walking Round the Outside
  3. Regular and Irregular
  4. Angle Sums With Letters
  5. Symmetry
  6. Building Polygons From Triangles
  7. Diagonals and Tilings

Unit 14: Angle Chasing

One angle leads to the next

  1. Following the Trail
  2. Angles in a Circle
  3. Chains of Isosceles Triangles
  4. Tangents Bring Right Angles
  5. Four Corners on a Circle
  6. Angles Between Shapes
  7. The Long Chase

Unit 15: Area

Covering space, and what it costs

  1. Rectangles and Halves
  2. Shapes Made of Shapes
  3. Units of Area
  4. Perimeter Is Not Area
  5. Areas on a Grid
  6. Cutting and Rearranging
  7. What Happens When You Scale

Unit 16: The Grid

Where geometry meets algebra

  1. Naming a Point
  2. How Far Apart
  3. Shapes on the Grid
  4. Reading a Straight Line
  5. Working Out the Shape
  6. Steepness on the Grid
  7. Lines and Their Steepness

Unit 17: Power of a Point

Lines cutting through circles

  1. Two Chords Crossing
  2. Cutting From Outside
  3. Chords and the Centre
  4. Two Tangents From One Point
  5. Inside, Outside, and On
  6. Circles Meeting Circles
  7. One Theorem All Along

Unit 18: Solid Shapes

Volume, and skin

  1. Stacking Slices
  2. Shapes That Taper
  3. Counting the Parts
  4. How Much It Holds
  5. Slicing and Unfolding
  6. Solids in Real Life
  7. Unfolding a Solid

Unit 19: Moving Shapes

Shifts, turns, flips and stretches

  1. Sliding and Flipping
  2. Turning
  3. Mirrors at an Angle
  4. One After Another
  5. Naming What Happened
  6. Enlarging From a Point
  7. Stretches and Squeezes

Unit 20: Mixed Geometry

Choosing the right tool

  1. Shaded Areas
  2. Special Triangles
  3. Shapes That Overlap
  4. One Shape Inside Another
  5. Ratios of Areas
  6. Two Ways to the Same Area
  7. Whatever It Takes

Unit 21: Machines

Functions, in and out

  1. In, Then Out
  2. What Goes In, What Comes Out
  3. Finding the Rule
  4. Drawing a Function
  5. Moving a Graph About
  6. Functions Made of Functions
  7. Undoing a Machine

Unit 22: Not Equal

Bigger, smaller and in between

  1. Reading the Signs
  2. The Sign That Turns Round
  3. Drawing the Answer
  4. More Than One Step
  5. Inequalities in Real Life
  6. Comparing Sizes
  7. Ranges and Distances

Unit 23: Making Up Rules

Operations, and how they behave

  1. Invent an Operation
  2. Brackets, Identities and Undoing
  3. More Invented Operations
  4. Reading an Operation Table
  5. Which Rules Survive
  6. Relations and Comparing
  7. Order, and Relations

Unit 24: Patterns in a Row

Sequences, and adding them up

  1. Same Step Every Time
  2. Gauss and the Schoolteacher
  3. The nth Term
  4. Sequences With Shapes
  5. Multiplying Sequences
  6. Sums Worth Knowing
  7. Adding Forever

Unit 25: Counting Cleverly

How many ways?

  1. Choices Multiply
  2. Putting Things in Order
  3. Listing Without Missing Any
  4. When Things Repeat
  5. Counting With Conditions
  6. Choosing and Splitting
  7. Pascal’s Triangle
  8. When Order Does Not Matter

Unit 26: Chance and Averages

Statistics and probability

  1. Three Kinds of Middle
  2. How Likely
  3. Data in a Table
  4. Listing Every Outcome
  5. Tree Diagrams
  6. What to Expect
  7. Is the Game Fair?
  8. Two Things at Once

Unit 27: Collections

Sets, overlaps and counting them

  1. A Bag of Things
  2. Joining and Overlapping
  3. Describing a Set
  4. Three Circles
  5. Sets and Chance
  6. Counting the Subsets
  7. Counting With Circles

Unit 28: Prove It

Being sure, not just fairly sure

  1. Examples Are Not Enough
  2. Letters Cover Every Case
  3. Making a Guess Worth Testing
  4. Proving Things About Multiples
  5. Turning a Statement Round
  6. Standing on Every Rung
  7. Two Great Proofs

Other things to learn

Start learning — units 1–2 are free