10th Grade Math Study Guide

Zoe and Max are fifteen — sophomores, and for the first time in ten years they are in two different math classes. American tenth grade splits four ways: Geometry on the standard track, Algebra II on the accelerated track, Integrated Mathematics II on the integrated track, and a second run at Algebra I for anyone who needs it. This thirty-three chapter read-aloud guide covers the union of Geometry, Algebra II, and Integrated Math II — with an opening chapter mapping every chapter to every course name and the 1892 Committee of Ten report that is the reason American high school math is sliced this way at all. Coverage is exhaustive: rigid motions as functions on the plane and congruence redefined as the existence of one, triangle congruence and similarity criteria derived rather than assumed, compass-and-straightedge constructions with their justifications, formal proof about lines and angles and quadrilaterals and polygons, the concurrency points and the Euler line, right-triangle trigonometry sharpened to exact values and identities, the Law of Sines and the Law of Cosines with the ambiguous case that gives zero, one, or two triangles, arcs and chords and the inscribed-angle theorem, tangents and secants and the power of a point, arc length and sector area and the birth of radian measure, cross-sections and solids of revolution, Euler's polyhedron formula and the proof that there are exactly five Platonic solids, volume by Cavalieri's principle including the sphere, compound and conditional probability with permutations, combinations, and expected value, complex number arithmetic and the complex plane and the fundamental theorem of algebra, function families and the transformations that move any graph, polynomial functions including the remainder, factor, rational-root, and binomial theorems, rational expressions with asymptotes and extraneous solutions, radical functions and composition and inverses, exponential growth and decay and the number e, logarithms and their laws proved from the exponent laws, arithmetic and geometric sequences and series (including the infinite geometric sum), the unit circle and radian trigonometry with the full graphs of amplitude, period, and phase, conic sections from the slice of a cone to the equation on the page, and statistical inference with sample surveys, observational studies, randomized experiments, and margin of error. Optional "Around the World Challenges" bring in plane vectors and the dot product, polar form of a complex number, and Chinese senior-one solid geometry; analytic geometry of the circle, trigonometric equations, and Israeli Bagrut inequality proofs; and surds, partial fractions, the R-formula, and a first taste of the gradient as a derivative from Singapore's Additional Mathematics. The two marquee "Story Corners" are the invention of the logarithm — sixteenth-century astronomers drowning in fifteen-digit multiplications, the trigonometric trick called prosthaphaeresis that turned products into sums, John Napier's twenty years of hand computation published in 1614, Henry Briggs riding three hundred miles to Edinburgh in 1615 to meet him, Jost Bürgi arriving independently and being beaten to print, Kepler using the tables to finish work he otherwise could not, the slide rule that carried the idea for 350 years and flew on Apollo 13, and the pocket calculator that ended it in 1972 — and the long homelessness of the complex numbers, from Caspar Wessel's 1799 Danish paper that correctly described the complex plane and was ignored for 96 years, to Jean-Robert Argand's 1806 Paris pamphlet that got the credit, to Gauss who had it first and printed it last (and thought the words "imaginary" and "negative" had done real damage), to William Rowan Hamilton in 1837 making the whole thing ordinary by defining a complex number as a pair of ordinary numbers. Along the way: the Erlangen Program's claim that a geometry is the study of what a group of motions leaves alone; Gauss at nineteen constructing the seventeen-sided polygon and asking for it on his tombstone; Marjorie Rice, the San Diego homemaker with a high school education who found four new pentagon tilings; Liu Hui and Zu Geng finishing the sphere volume around 500 CE, eleven centuries before Cavalieri; Cicero hunting through brambles in 75 BCE for the tomb of Archimedes and finding it by the carving of a sphere inside a cylinder; the Monty Hall problem that drew ten thousand angry letters (a thousand from PhDs, almost all wrong); Abel and Galois printing the impossibility of the quintic and being ignored; Oresme's four-line proof that a series can add to infinity even as its terms shrink to nothing; Ptolemy's theorem as the true origin of the angle-sum formulas; Fourier claiming every wave is a sum of sines and having his memoir held back for fifteen years; a brass tide-predicting machine that added sine waves with pulleys; a Guinness brewer publishing the most-used test in statistics under the false name "Student"; and the 1954 polio field trial that ran two designs on 1.8 million children and settled, permanently, why randomization is not optional. Complete practice with full answer keys. A read-aloud math book that teaches not just how, but why. Book ten of the Zoe & Max grades 1–12 math series.



Al-generated content disclosure: This work was created with the help and usage of artificial intelligence. All sources are cited in the bibliography section.