Numbers

Numbers are the oldest tool in mathematics. People first counted with marks on sticks; later they found ways to write huge numbers, share things into parts, and even name numbers that can never be written out in full. This ladder climbs from counting tricks to the complex numbers, where \(-1\) finally gets a square root.

8 levels, 53 lessons. The ages are only a rough guide for parents and teachers; anyone can start at any level.

Level	Rough age	Lessons
Level 1: First numbers	5-6	Counting things; Seeing small amounts at once; More, fewer, the same; Putting together and taking away; Making ten; Teens and beyond
Level 2: Tens, hundreds and sharing	7-8	Place value to 1000; Adding and subtracting two- and three-digit numbers; Fact families and mental tricks; Multiplying as equal groups; Dividing as fair sharing and as making groups; Halves, thirds and quarters; Old ways to write numbers
Level 3: Big numbers, times tables and fractions	9-10	Numbers to a million and beyond; Times tables with understanding; Multiplying bigger numbers; Division with remainders; Fractions are numbers; Equal and comparing fractions; Decimals: tenths and hundredths
Level 4: Working with fractions and decimals	11-12	Adding and subtracting fractions; Multiplying fractions; Dividing fractions; Calculating with decimals; Fractions to decimals and back; Negative numbers; Order of operations
Level 5: Integers, powers and roots	12-13	Multiplying and dividing integers; Rational numbers; Powers; Zero and negative exponents; Scientific notation; Square roots and cube roots; Other bases: binary, base 5 and hexadecimal
Level 6: The real numbers	13-14	Repeating decimals to fractions; Rational and irrational numbers; Proof that √2 is irrational; Radicals; The real numbers; Estimation and precision; Approximating roots
Level 7: Building number systems	15-16	Why each family of numbers was invented; The number i; Complex numbers a + bi; The complex plane; Every quadratic has roots; Intervals and absolute value
Level 8: Complex numbers and beyond	17 and beyond	Polar form; De Moivre's theorem; Roots of unity; Euler's formula; The fundamental theorem of algebra; Beyond the complex numbers