Khan Academy’s math program is designed to help children and young adults learn and practice a particular set of math skills. These skills map to school curricula like the US Common Core, so that when students learn something on the site, it translates to success in the classroom. But Khan Academy can also help adults review the fundamental skills necessary to learn more advanced math. The idea is to fill in “swiss cheese gaps” in knowledge that often accumulate when learning math in school. These gaps can slow down further learning, since math success in later courses depends on knowing the skills taught in previous courses.

This week, I’ll go over the topics available on Khan Academy. Next week, I’ll look at how they relate to discrete math topics.

## Courses

Khan Academy is in the middle of a mastery system upgrade. The previous World of Math system coexists with a new mastery system based on courses. Fortunately, the two systems share mastery status for each skill. So if you use one system for practice, you get credit in the other system and don’t have to review topics you already know.

I find the World of Math system simpler to navigate: you can just keep clicking the “Start mastery challenge” button, and problems keep appearing. You don’t have to worry about which subjects the problems come from. But this flat structure makes it harder to see how problems fit into the overall math curriculum. Since I’m writing about curriculum this week, I’ll use the new system’s course-based structure.

Khan Academy currently offers 57 math courses, but many topics are covered by multiple courses. For example, the Differential Calculus, AP®︎ Calculus AB, and Calculus 1 courses each work as a first course in Calculus. The World of Math mission draws from many of the 57 courses, but it picks a subset of the exercises to avoid duplication.

Here’s how I would organize these 57 courses into 15 topic areas by consolidating courses that cover similar topics:

**Early math (through 2nd grade)**

This is what kids learn in their preschool years and the first three years of primary school. Topics include counting, addition and subtraction, place value, measurement and data, and basic geometry.

4 courses: Early Math, Kindergarten, 1st grade, and 2nd grade.

**Primary school (3rd-5th grades)**

These courses review and expand on the early math topics, and add multiplication and division, fractions and decimals, rounding, calculating area, word problems, negative numbers, factors, unit conversion, and algebraic thinking.

11 courses: 3rd grade, 3rd grade (Eureka Math/EngageNY), 3rd grade foundations (Eureka Math/EngageNY), Arithmetic, Arithmetic (all content), 4th grade, 4th grade (Eureka Math/EngageNY), 4th grade foundations (Eureka Math/EngageNY), 5th grade, 5th grade (Eureka Math/EngageNY), and 5th grade foundations (Eureka Math/EngageNY).

**Later primary school (6th-7th grades)**

These courses add more advanced topics to prepare students for later studies: ratios and proportional relationships, rates, percentages, properties of numbers, variables, expressions, equations, inequalities, probability and statistics, ratios, rational numbers, and calculating volume.

8 courses: 6th grade, 6th grade (Illustrative Mathematics), 6th grade (Eureka Math/EngageNY), 6th grade foundations (Eureka Math/EngageNY), 7th grade (Eureka Math/EngageNY), 7th grade, 7th grade (Illustrative Mathematics), and 7th grade foundations (Eureka Math/EngageNY).

**Pre-algebra and geometry (8th grade)**

The 8th grade courses combine pre-algebra and basic geometry topics, to prepare students for full courses on those subjects in high school:

- Pre-algebra: numbers and operations, solving equations with one unknown, linear equations and functions, systems of equations, integer exponents, scientific notation, exponents, radicals, arithmetic properties, factors and multiples, and the coordinate plane.
- Basic geometry: transformations, congruence, similarity, Pythagorean theorem, and irrational numbers.

6 courses: 8th grade, 8th grade (Illustrative Mathematics), 8th grade (Eureka Math/EngageNY), 8th grade foundations (Eureka Math/EngageNY), Pre-algebra, and Basic geometry.

**Algebra**

A first course in high-school algebra, covering algebraic expressions, solving linear equations and inequalities, graphing lines, slope, systems of equations, expressions with exponents, quadratics and polynomials, functions, linear word problems, sequences, systems of equations, absolute value, piecewise functions, rational exponents, radicals, exponential growth/decay, polynomials, factoring, exponential and logarithmic functions, quadratic functions, complex numbers, conic sections, vectors, and matrices.

4 courses: Algebra basics, Algebra I, Algebra (all content), and Algebra I (Eureka Math/EngageNY).

**Geometry**

A first course in high-school geometry, covering lines, angles, shapes, triangles, quadrilaterals, circles, the coordinate plane, area and perimeter, volume and surface area, the Pythagorean theorem, transformations, congruence, similarity, right triangles, trigonometry, analytic geometry, solid geometry, constructions, and proofs.

3 courses: High school geometry, Geometry (Eureka Math/EngageNY), and Geometry (all content).

**Algebra II**

A second course in high-school algebra, covering functions, complex numbers, polynomial arithmetic, radical relationships, rational relationships, exponential growth and decay, exponentials and logarithms, trigonometry, advanced equations and functions, series, conic sections.

2 courses: Algebra II and Algebra II (Eureka Math/EngageNY).

**Trigonometry**

Trigonometry for right triangles and general triangles; the unit circle definition of sin, cos, and tan; graphs of trigonometric functions; trigonometric equations and identities.

1 course: Trigonometry.

**Precalculus**

Everything you need to know before you study calculus, assuming you mastered the previous courses.

Topics: trigonometry, conic sections, vectors, matrices, complex numbers, probability, statistics, combinatorics, series, rational and exponential functions.

2 courses: Precalculus and Precalculus (Eureka Math/EngageNY).

Also, these three courses cover the high school curriculum through precalculus: Mathematics I, Mathematics II, and Mathematics III.

**Calculus**

The traditional pinnacle of high school math, which the previous courses lead up to. Includes courses to prepare for the two AP®︎ Calculus tests.

Topics: Limits, continuity, derivatives, techniques for differentiation, applications of derivatives, analyzing functions, parametric equations, polar coordinates, vector-valued functions, integrals, differential equations, applications of integrals, infinite sequences and series.

6 courses: Differential Calculus, Integral Calculus, Calculus 1, Calculus 2, AP®︎ Calculus AB, and AP®︎ Calculus BC.

**Multivariable calculus**

Post-high school calculus.

Derivatives of multivariable functions, applications of multivariable derivatives, integrating multivariable functions, Green’s, Stokes’, and the divergence theorems.

1 course: Multivariable calculus.

**Differential equations**

First order differential equations, second order linear equations, Laplace transform.

1 course: Differential equations.

**Linear algebra**

Vectors and spaces, matrix transformations, alternate coordinate systems (bases).

1 course: Linear algebra.

**Probability and statistics**

Khan Academy statistics coverage is thorough, and includes a course to prepare for the AP®︎ Statistics exam.

Topics: Advanced regression (inference and transforming); analysis of variance (ANOVA); analyzing categorical data; chi-square tests for categorical data; confidence intervals; counting, permutations, and combinations; displaying and comparing quantitative data; displaying and describing quantitative data; exploring bivariate numerical data; inference comparing two groups or populations; modeling data distributions; probability; random variables; sampling distributions; significance tests (hypothesis testing); study design; summarizing quantitative data; two-sample inference for the difference between groups.

3 courses: Statistics and probability, High school statistics, and AP®︎ Statistics.

**Competitive and recreational math**

This section includes recreational math topics, plus courses that cover the two tests used to determine qualification for the United States of America Mathematical Olympiad (USAMO): 1) the American Mathematics Competition (AMC) and 2) the American Invitational Mathematics Examination (AIME) (AIME).

1 course: Math for fun and glory.

Khan Academy’s math curriculum is a work in progress. New skills periodically show up in the World of Math and in the course system. But there’s already plenty available to ensure that your fundamental math skills are well-polished.

*I’m writing about discrete math and competitive programming this year. For an introduction, see A Project for 2019. To read the whole series, see my Discrete Math category page.*