The order, stage by stage
These are the 6 chapters of school maths on our map, from counting up to the material just before calculus, with the number of concepts in each and an estimate of the hours to learn them from nothing.
- Foundations of Arithmetic: 155 concepts, about 39 hours.
- Pre-Algebra: 92 concepts, about 31 hours. Builds directly on Foundations of Arithmetic.
- Algebra I: 202 concepts, about 97 hours. Builds directly on Foundations of Arithmetic and Pre-Algebra.
- Geometry: 269 concepts, about 126 hours. Builds directly on Foundations of Arithmetic, Pre-Algebra and Algebra I.
- Trigonometry: 102 concepts, about 52 hours. Builds directly on Foundations of Arithmetic, Pre-Algebra, Algebra I and Geometry.
- Pre-Calculus & Discrete Foundations: 79 concepts, about 41 hours. Builds directly on Foundations of Arithmetic, Pre-Algebra, Algebra I, Geometry and Trigonometry.
Together that is 899 concepts and about 385 hours, roughly 18 months at 5 hours a week. Geometry is the biggest stage by hours. Each stage builds on the ones before it, so a topic you skip turns up again as a step inside later ones. The order is not perfectly strict: 6 of the 899 concepts lean on a later stage, for example complex conjugate root theorem (in Algebra I, which needs Pre-Calculus & Discrete Foundations) and Stewart's theorem (in Geometry, which needs Trigonometry). Those can wait until you reach the later stage.
How the hours are worked out: every concept on the map has a difficulty level, and each level gets a fixed time for the lesson, practice and a first review (L1 15 minutes, L2 20 minutes, L3 30 minutes, L4 40 minutes and L5 50 minutes). Those times are our own estimate, not an average measured from learners, and they leave out time spent relearning anything you forget along the way.
Calculus comes after the last stage. If that is where you are heading, how long it takes to learn calculus counts that route and shows which of these stages calculus does not need in full.
Finding your own starting point
If you went to school, you are probably not starting from nothing. More often some stages are in good shape, some are partly there, and there are a few specific gaps. The useful question is where your secure knowledge stops. Three questions give a rough first answer:
- Which is bigger, 5/8 or 3/5? (Over a common denominator of 40 they are 25/40 and 24/40, so 5/8.) If this one is slow, start in the fractions part of the first stage.
- Solve 3(x − 4) = 2x + 1. (3x − 12 = 2x + 1, so x = 13.) If expanding the bracket or moving terms across is shaky, start with linear equations in the second stage.
- Expand (x + 3)(x − 5). (x² − 2x − 15.) If that is unfamiliar, the third stage is your starting point.
Three questions are a rough guide, not a measurement. A stage can feel solid while one concept inside it is missing, and on the map a missing concept low down sits under far more than its own chapter. For instance, equivalent fractions is a direct prerequisite of equivalent ratios: writing 2/3 as 10/15 is the same move as writing the ratio 2:3 as 10:15, so a gap in the fraction step comes back as trouble with ratios.
What books and videos are good for
A good textbook or video series can explain each topic well. What it cannot do is tell which of its chapters you already know, so you either start at page one and re-cover what you have, or guess where to jump in. It also will not bring a topic back a few weeks later to check it has stuck. Use explanations for explaining, and use something else to decide where to start and when to review: a short diagnostic test, and a review schedule, whether on paper or in software. For how to use a study session once you are under way, see how to study maths.
If a bad exam result is what makes starting again hard, failed maths? now what looks at what that result did and did not measure. The age question is answered in the questions below; sources checked on 27 October 2026, including Hartshorne and Germine (2015).
