Factors & Multiples (Classical)

Classical mathematics teaches a child to look at numbers and ask what they are made of. This Grade 4 program covers factors and multiples in the classical tradition: definitions stated formally and learned by heart, Latin and Greek roots explained, and systematic practice that turns each idea into a habit before the next is introduced. There is no story dressed up to disguise the math. The math is the point, and the point is worth making plainly.

Week 1 begins with the word itself. Factor comes from the Latin facere — to make. A factor is one of the whole numbers that, multiplied together, makes a product. Children list factor pairs for numbers up to 50 and meet the idea of divisibility — that a factor divides its number evenly, with nothing left behind. Week 2 introduces prime and composite numbers. The Greeks called primes protos arithmos, the first numbers, because they cannot be built from smaller whole-number factors. Children use the Sieve of Eratosthenes to find every prime to 100 and learn why 1 is set apart from both categories.

Week 3 turns the work into mental math. The classical divisibility rules for 2, 3, 4, 5, 6, 9, and 10 are drilled until they are automatic — and each rule is grounded in the structure of place value, not memorized as a trick. Week 4 introduces the Greatest Common Factor by the steady method of listing every factor of two numbers, marking the common ones, and choosing the greatest. Week 5 pairs GCF with the Least Common Multiple. Where factors are few, multiples march on without end, so the LCM is the smallest of the shared ones.

The second half of the program builds upward. Week 6 teaches prime factorization through the factor tree, with the Fundamental Theorem of Arithmetic stated outright: every whole number greater than 1 has exactly one prime form. Week 7 connects multiples to patterns — the multiples of 5 always end in 5 or 0; perfect squares have an odd number of factors because the middle factor pairs with itself. Week 8 generates number sequences from a rule and asks the harder question: what features does the pattern carry that the rule never named? Week 9 is a cumulative capstone covering every preceding week.

Each week contains five worksheets of ten problems, progressing from definition recall through systematic drill to analytical reasoning. Answer keys explain the work in full, so a parent who hasn’t done this math in twenty years can guide the discussion with confidence. The classical method is patient and exact, and a child who has worked through these nine weeks will see numbers differently afterward.