Geometry: free lessons and course outline

Geometry is the one high school mathematics course that is mostly about justification rather than calculation, and that is why capable algebra students so often find it hard. The question stops being what is the answer and becomes how do you know, and a student who has never been asked that before has to learn a new habit from scratch. This course is built around that habit. Every claim in every worked example names the definition, postulate or theorem that licenses it, and no step is ever justified by how the diagram looks. The eleven units follow the California traditional pathway in the order the standards intend: the undefined terms and the constructions that make them concrete, then reasoning and proof as a subject in its own right, parallel lines, transformations and the definition of congruence through rigid motion, congruent triangles, the special segments and inequalities inside a triangle, quadrilaterals and their hierarchy, similarity and dilation, right-triangle trigonometry, circles, and the area and volume work the standards close on. The four errors that cost geometry students the most marks are named where they arise: reading a fact off the picture, using the theorem you are trying to prove, confusing a statement with its converse, and writing a congruence statement whose letters do not correspond. Every lesson ends with ten problems, every unit with a ten-problem review, and the year with six pieces of mathematical writing that have full model responses.