Pre-Algebra (Classical)

The medieval universities counted mathematics among the four liberal arts of the quadrivium — arithmetic, geometry, music, astronomy — disciplines a free person studied because they trained the mind toward truths that did not depend on opinion. Pre-algebra belongs in that tradition. In the classical approach, etymology, history, and proof are not decorative add-ons; they are how the subject is taught. A student who learns that exponent comes from Latin exponere — “to set forth” — and that the small superscript was standardized only in 1637, when Descartes published La Géométrie, holds the notation differently than one who has merely memorized a rule.

Week 1 derives the three laws of exponents — product, quotient, power — from the definition rather than handing them down. Week 2 extends them to zero and negative powers, showing that x⁰ = 1 and x⁻ⁿ = 1/xⁿ are not arbitrary decrees but the only definitions that keep the laws consistent.

Scientific notation, by way of Archimedes

Weeks 3 and 4 take up scientific notation. The historical passage carries students to Psammites — The Sand Reckoner — written by Archimedes of Syracuse around 250 BC, who counted the grains that would fill the universe using an “orders of myriads” system and arrived at roughly 10⁶³. Week 4 teaches the four operations on numbers in this form, built around Avogadro’s number 6.022 × 10²³ — not actually measured until Jean Perrin’s 1909 experiment.

The radical, and the scandal of the Pythagoreans

Week 5 introduces the symbol √ — from Latin radix, “root,” which also gives English the radish and eradicate. The mark first appeared in print in 1525, in Christoff Rudolff’s Coss, probably as a stylized lowercase r. Students memorize the perfect squares to 15² and cubes to 10³, then walk through the proof by contradiction that √2 is irrational. The Pythagoreans had built their philosophy on the conviction that all quantity is rational — and the diagonal of their own unit square broke it. Legend has Hippasus drowned at sea for revealing it.

Week 6 is Pythagoras at Croton c. 530 BC and a full geometric proof of a² + b² = c² by rearrangement of areas — four right triangles inside a larger square, area computed two ways, algebra reducing to the theorem. Babylonian tablets had listed Pythagorean triples twelve centuries earlier; the Greeks were first to prove the relation must hold universally. Week 7 derives the distance formula from that theorem in the Cartesian plane Descartes published in the same 1637 appendix. Applications: the surveyor crossing a marsh, the carpenter’s 3-4-5 trick, the three-dimensional diagonal.

Weeks 8 and 9 are synthesis and capstone. The closing case study uses Newton’s F = G · m₁m₂ / r² from the Principia of 1687 to compute the force between two solar-mass stars at 3 × 10¹¹ meters — pulling on Week 1’s exponents, Week 2’s negative exponent inside G, every operation from Weeks 3-4, and Week 5’s squaring. Students aren’t applying memorized rules by the end; they’re deriving them.

The word algebra itself descends from the Arabic al-jabr in al-Khwarizmi’s 9th-century treatise — and from his name, Latinized, comes algorithm. Mathematics is a Greek, Latin, and Arabic inheritance, and these worksheets treat it as such. Five practice sets per week, full answer keys, print-ready.