Appendix D: The US Mathematics Curriculum
If you were educated in the UK, the American system can be baffling: maths is called "math", it is taught as a sequence of separately named year-long courses rather than as one two-year syllabus, and a fifteen-year-old can be "doing Algebra II" or "in AP Calc" depending on which track they are on. This appendix is a translation guide. It explains the structure, compares it with GCSE and A-level, and maps each US course to the chapters of this book that cover it.
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Read this as a cross-reference, not a rulebook
The United States has no national curriculum. Each of the 50 states sets its own standards, each of roughly 13,000 school districts chooses its own courses, textbooks and pacing, and individual schools offer different tracks. Everything below describes the typical pattern. Course names, grade levels and content vary from place to place; when it matters (for example, when choosing a course for next year), check your school’s own course catalogue. |
D.1. Grades, ages, and school levels
US school years are called grades. A child usually starts kindergarten (K) in the year they turn 5 or 6 (the cut-off date varies by state) and finishes 12th grade at 17 or 18. Because the English Reception year comes before Year 1, US grade \(n\) corresponds roughly to English Year \(n + 1\).
| US grade | Typical age during the year | School level | Rough English equivalent |
|---|---|---|---|
Pre-K |
4-5 |
Preschool |
Reception |
K |
5-6 |
Elementary school |
Year 1 |
1-5 |
6-11 |
Elementary school |
Years 2-6 (KS1-KS2) |
6 |
11-12 |
Middle school (sometimes elementary) |
Year 7 |
7-8 |
12-14 |
Middle school or junior high |
Years 8-9 (KS3) |
9 ("freshman") |
14-15 |
High school |
Year 10 |
10 ("sophomore") |
15-16 |
High school |
Year 11 (GCSE exams) |
11 ("junior") |
16-17 |
High school |
Year 12 (Lower Sixth) |
12 ("senior") |
17-18 |
High school |
Year 13 (Upper Sixth; A-level exams) |
So a US 15-year-old is usually in 9th or 10th grade, the equivalent of the two GCSE years.
D.2. How US high-school mathematics is organised
The biggest difference from the UK is structural.
Courses, not syllabuses. In England, mathematics from 14 to 16 is one two-year GCSE course ending in national exams, and from 16 to 18 one two-year A-level course ending in national exams. In the US, each school year is a separately named course (Algebra I, Geometry, Algebra II, …), each lasting one year, each graded by the teacher, and each recorded on the student’s transcript.
Continuous assessment. The course grade (A to F) is built from homework, quizzes, unit tests and a final exam set by the school. These grades feed into a grade point average (GPA), which universities weigh heavily. There is no national exit exam in mathematics, although many states set end-of-course (EOC) tests, especially for Algebra I.
Breadth until 18. US students keep studying six or seven subjects until they graduate. There is no equivalent of choosing three or four A-levels at 16. High schools typically require three or four years of mathematics to graduate (commonly including Algebra I and Geometry, often Algebra II), and selective universities expect four years, ideally reaching Precalculus or Calculus.
Tracks. Students in the same grade can be in very different courses. Placement into an accelerated track usually happens around grades 6-7 and determines how far a student gets by 12th grade.
D.3. The typical course sequence
| Grade | Standard track | Accelerated track | Highly accelerated |
|---|---|---|---|
6 |
Math 6 |
Math 6 (advanced) |
Accelerated Math 6 |
7 |
Math 7 |
Accelerated Math 7 / Pre-Algebra |
Pre-Algebra or Algebra I |
8 |
Math 8 / Pre-Algebra |
Algebra I |
Algebra I or Geometry |
9 |
Algebra I |
Geometry (Honors) |
Geometry or Algebra II (Honors) |
10 |
Geometry |
Algebra II (Honors) |
Algebra II or Precalculus (Honors) |
11 |
Algebra II |
Precalculus (Honors or AP) |
AP Calculus BC |
12 |
Precalculus, Statistics, or another elective |
AP Calculus AB or BC; AP Statistics |
Multivariable Calculus, Linear Algebra (often via dual enrollment); AP Statistics |
The "standard" route is the classic sequence
and the tracks differ mainly in when a student starts it. Taking Algebra I in 8th grade (rather than 9th) is the usual gateway to reaching calculus in high school.
Integrated pathways. A minority of states and districts teach Integrated Math I, II, III (sometimes called Math 1-3 or Secondary Math I-III) instead of Algebra I / Geometry / Algebra II. Each year mixes algebra, geometry, functions and statistics, which is closer to the UK approach. The total content over three years is roughly the same.
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Why does Geometry get a whole year?
The US "layer-cake" sequence, with a year of Euclidean geometry sandwiched between two years of algebra, dates back to curriculum recommendations made in the 1890s, when Euclid was still the model of rigorous thinking. It survives today, and it has an upside: the US Geometry course is where most American students first write formal proofs, often in a distinctive two-column format (statement on the left, reason on the right). A typical UK student meets far less formal proof before A-level. |
D.4. Common Core, in brief
The Common Core State Standards for Mathematics (2010) are a set of grade-by-grade expectations that most states adopted in the early 2010s. A few states never adopted them (Texas and Virginia, for example, have their own standards), and several have since revised or renamed them, but most state standards still closely resemble Common Core.
Key points for a parent:
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They are standards (what a student should be able to do by the end of each grade), not a curriculum or a textbook. Districts choose their own materials.
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Grades K-8 are specified year by year. High school is specified by topic area (Number and Quantity, Algebra, Functions, Modeling, Geometry, Statistics and Probability), so the same standards can be delivered as the traditional sequence or as an integrated one.
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Standards marked (+) are extra topics for students heading towards calculus and STEM subjects, such as complex-number arithmetic, vectors and matrices.
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Eight Standards for Mathematical Practice run through every grade: make sense of problems and persevere; reason abstractly and quantitatively; construct viable arguments; model with mathematics; use tools strategically; attend to precision; look for and use structure; look for regularity in repeated reasoning.
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What parents most often notice is the emphasis on multiple strategies and representations (number lines, area models for multiplication, "explain your reasoning"). That emphasis is close in spirit to this book’s insistence on connecting words, symbols, pictures and computation.
D.5. Honors, AP, and IB
Honors. An Honors course covers the same course name at greater depth and pace, with harder problems. Many schools give Honors and AP grades extra weight in the GPA.
Advanced Placement (AP). AP courses follow a national syllabus set by the College Board and end in a standardised exam in May, scored 1 to 5. Many universities give credit or advanced placement for a 4 or 5 (some for a 3). AP is the closest US analogue to an A-level: a national, externally assessed exam in one subject. The AP mathematics courses are:
| Course | Content | Chapters |
|---|---|---|
AP Precalculus |
Polynomial and rational functions; exponential and logarithmic functions; trigonometric and polar functions. (Parametric functions, vectors and matrices appear in the course but not on the exam.) |
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AP Calculus AB |
Limits and continuity; derivatives and their rules; implicit differentiation; related rates; L’Hôpital’s rule; optimisation and curve sketching; Riemann sums; definite and indefinite integrals; the Fundamental Theorem; integration by substitution; slope fields and separable differential equations; area and volume. Roughly a first-semester university calculus course. |
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AP Calculus BC |
Everything in AB, plus integration by parts and partial fractions; improper integrals; Euler’s method; logistic growth; parametric, polar and vector-valued functions; infinite series (convergence tests, Taylor and Maclaurin series). Roughly two semesters of university calculus. The BC exam also reports an AB sub-score. |
As AB, plus Chapter 14, Chapter 32 |
AP Statistics |
Exploring and describing data; sampling and experimental design; probability; random variables; binomial and geometric distributions; the normal distribution; sampling distributions; confidence intervals and significance tests (means, proportions, chi-square, regression slope). Roughly an introductory, non-calculus university statistics course. |
Related AP courses that use mathematics heavily: AP Computer Science A (Java programming), AP Computer Science Principles, AP Physics 1 (algebra-based mechanics) and AP Physics C: Mechanics (calculus-based mechanics).
International Baccalaureate (IB). Some US high schools offer the two-year IB Diploma in grades 11-12. Its mathematics comes in two flavours, Analysis and Approaches (more pure, calculus-heavy) and Applications and Interpretation (more statistics and modelling), each at Standard Level (SL) or Higher Level (HL). IB Analysis and Approaches HL is roughly comparable to A-level Mathematics plus some Further Mathematics.
Dual enrollment. High-school students can take courses at a local community college or university, earning credit towards both their high-school diploma and a university degree. This is how strong students continue after AP Calculus BC: Multivariable Calculus, Linear Algebra, Differential Equations and Discrete Mathematics are the usual next courses.
D.6. SAT and ACT mathematics
Most US universities consider (and some require) a standardised admissions test, the SAT or the ACT. Both include mathematics, and both test the content of Algebra I, Geometry and parts of Algebra II, with no calculus. Their formats change from time to time, so check the current specifications, but at the time of writing:
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SAT Math (College Board): a computer-adaptive section in two modules, 44 questions in 70 minutes, scored 200-800. A calculator (including a built-in graphing calculator) is allowed throughout. Four content areas: Algebra (linear equations, inequalities and systems); Advanced Math (quadratics, polynomials, exponentials, function notation); Problem-Solving and Data Analysis (ratios, percentages, units, statistics, probability); Geometry and Trigonometry. Relevant chapters: Chapter 2 to Chapter 10, the start of Chapter 13, and the descriptive parts of Chapter 31.
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ACT Math: 45 multiple-choice questions in 50 minutes, scored 1-36, calculator allowed. Slightly broader than the SAT, with occasional logarithms, matrices, complex numbers and trigonometry from Algebra II and Precalculus.
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The PSAT/NMSQT, taken in 10th or 11th grade, is a shorter version of the SAT and the qualifying test for National Merit Scholarships.
D.7. The UK and US systems side by side
| UK stage | Age | Closest US counterpart | Chapters |
|---|---|---|---|
Key Stage 3 (Years 7-9) |
11-14 |
Math 6, 7, 8 and Pre-Algebra; Algebra I for accelerated students |
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GCSE Mathematics, Foundation tier |
14-16 |
Pre-Algebra and Algebra I, with basic geometry and statistics |
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GCSE Mathematics, Higher tier |
14-16 |
Algebra I plus Geometry plus parts of Algebra II: quadratics, simultaneous equations, functions, right-triangle trigonometry, sine and cosine rules, circle theorems, basic vectors, probability. No calculus beyond informal gradients and areas under graphs |
Chapter 4 to Chapter 10, start of Chapter 13, Chapter 29 |
Level 2 Further Maths / Additional Maths |
15-16 |
Honors Algebra II and early Precalculus |
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A-level Mathematics (pure) |
16-18 |
Algebra II, Precalculus and most of AP Calculus BC: proof, functions, logs and exponentials, trigonometric identities, sequences and series, differentiation and integration (including by parts and substitution), parametric equations, simple differential equations, numerical methods (Newton-Raphson, trapezium rule) |
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A-level Mathematics (statistics) |
16-18 |
Much of AP Statistics: sampling, binomial and normal distributions, hypothesis testing, correlation and regression |
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A-level Mathematics (mechanics) |
16-18 |
Taught in physics in the US: AP Physics 1, or AP Physics C: Mechanics for the calculus-based version |
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A-level Further Mathematics |
16-18 |
Beyond the AP courses: complex numbers (including De Moivre), matrices, proof by induction, polar coordinates, hyperbolic functions, series, further calculus and differential equations. Comparable to first-year university Linear Algebra and Differential Equations, reached in the US by dual enrollment |
Some broad differences are worth keeping in mind.
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Specialisation. An A-level student studies mathematics for a large fraction of the school week from 16. A US student spends one period a day on it. In compensation, the US student usually also studies English, history, a science and a language to 18.
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Calculus timing. An A-level student meets calculus at 16-17. A US student on the standard track may not meet it until university; an accelerated student meets it at 16-17, much like the A-level student.
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Proof and geometry. US students meet formal two-column proof earlier (in Geometry, typically at 14-15). UK students meet more algebraic manipulation earlier.
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Statistics. Both systems now include it, but in the US it is often a separate course (AP Statistics) rather than a strand woven through every year.
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What happened to "Maths with Mechanics"?
In the 1980s and early 1990s many English A-level students took Mathematics with Mechanics, or Pure and Applied Mathematics: pure mathematics (algebra, trigonometry, calculus) alongside applied mathematics, which in practice meant Newtonian mechanics (kinematics, forces, Newton’s laws, projectiles, moments, energy). The US has no single course like it. The pure half corresponds to Algebra II, Precalculus and AP Calculus; the mechanics half lives in the physics department, where AP Physics C: Mechanics is the closest match because it uses calculus. |
D.8. Terminology: UK and US words for the same things
| UK | US | Notes |
|---|---|---|
maths |
math |
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Year 11 |
10th grade |
Subtract one from the English year number |
BIDMAS / BODMAS |
PEMDAS |
US mnemonic "Please Excuse My Dear Aunt Sally": Parentheses, Exponents, Multiplication/Division, Addition/Subtraction |
brackets |
parentheses |
US "brackets" usually means square brackets \(\lbrack\ \rbrack\) |
highest common factor (HCF) |
greatest common factor (GCF) or greatest common divisor (GCD) |
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index, indices, powers |
exponent(s) |
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surd |
radical |
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standard form (\(3.2 \times 10^5\)) |
scientific notation |
US "standard form" means a line written as \(Ax + By = C\), or a number written out in full |
\(y = mx + c\) |
\(y = mx + b\) |
The \(y\)-intercept is called \(b\) in the US |
gradient (of a line) |
slope |
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simultaneous equations |
system of equations |
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recurring decimal |
repeating decimal |
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modulus \(\lvert x \rvert\) |
absolute value |
"Modulus" is still used for complex numbers |
trapezium |
trapezoid |
Same shape: a quadrilateral with (at least) one pair of parallel sides. Confusingly, in older usage each country’s other word meant a quadrilateral with no parallel sides |
turning point |
vertex (of a parabola), extremum |
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stationary point |
critical point |
US "critical point" also includes points where the derivative is undefined |
sine rule, cosine rule |
law of sines, law of cosines |
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Pythagoras' theorem |
the Pythagorean theorem |
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Newton-Raphson |
Newton’s method |
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trapezium rule |
trapezoidal rule |
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sum to infinity |
sum of an infinite (geometric) series |
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differentiation from first principles |
using the limit definition of the derivative |
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nought |
zero |
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marks |
points |
D.9. Course-by-course map to this book
| Course | Typical grade (standard / accelerated) | Key topics | Chapters |
|---|---|---|---|
Elementary math |
K-5 |
Place value, the four operations, multiplication tables, fractions (from grade 3), decimals, measurement, simple geometry, simple data displays |
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Math 6 and Math 7 |
6-7 |
Ratios, rates and percentages; negative numbers; expressions; one-variable equations and inequalities; area, surface area and volume; introductory statistics and probability |
Chapter 1, Chapter 2, Chapter 4, start of Chapter 5, Chapter 9 |
Math 8 / Pre-Algebra |
8 / 7 |
Exponents and scientific notation; square roots and irrational numbers; linear equations and functions; introductory systems; the Pythagorean theorem; transformations |
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Algebra I |
9 / 8 |
Linear equations and inequalities; slope and lines; functions; systems; exponent rules; exponential functions (introduction); adding, multiplying and factoring polynomials; quadratics; data and scatter plots |
Chapter 4, Chapter 5, Chapter 6, start of Chapter 7, Chapter 8, Chapter 11 |
Geometry |
10 / 9 |
Congruence, similarity and transformations; two-column proof; right-triangle trigonometry; circles; coordinate geometry; area and volume |
Chapter 9, Chapter 10, start of Chapter 13, Chapter 22 |
Algebra II |
11 / 10 |
Polynomial and rational functions; radicals; complex numbers; exponential and logarithmic functions; sequences and series; inverse functions and transformations; often the unit circle and trigonometric functions; probability and statistics |
Chapter 7, Chapter 11, Chapter 12, Chapter 15, Chapter 14, Chapter 13, Chapter 29 |
Precalculus (including AP Precalculus) |
12 / 11 |
Functions in depth; trigonometric functions, identities and equations; polar coordinates and complex numbers in polar form; vectors; sometimes matrices and conic sections; sequences, series and an introduction to limits |
Chapter 11, Chapter 13, Chapter 15, Chapter 10, Chapter 14, Chapter 26, Chapter 27, Chapter 16 |
AP Calculus AB |
12 / 11 |
See the AP table above |
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AP Calculus BC |
12 / 11 |
AB plus series, parametric and polar calculus, further integration |
As AB, plus Chapter 14, Chapter 32 |
AP Statistics (or non-AP Statistics) |
11-12 |
Data, sampling and experiments, probability, distributions, inference |
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Discrete Mathematics (occasional high-school elective; standard at university) |
12 / university |
Logic, sets, proof and induction, number theory, counting, graphs |
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Multivariable Calculus (Calculus III) |
post-BC / university |
Partial derivatives, gradients, optimisation in several variables, multiple integrals, vector calculus |
Chapter 20 (the book stops before multiple integrals and vector calculus) |
Linear Algebra |
post-BC / university |
Vectors, matrices, systems, determinants, vector spaces, eigenvalues, orthogonality, the SVD |
D.10. Quick reference: what does "He’s doing …" mean?
- "She’s in Math 7"
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Ratios, percentages, negative numbers, simple equations. UK Year 8. Chapters Chapter 2, Chapter 4.
- "He’s doing Pre-Algebra"
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The bridge from arithmetic to symbols: exponents, roots, simplifying expressions, solving linear equations, the Pythagorean theorem. UK Years 8-9. Chapters Chapter 3, Chapter 4, Chapter 5.
- "She’s taking Algebra I in 8th grade"
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She is on an accelerated track, one year ahead of the standard sequence, and on course for calculus in high school. Content: lines, systems, functions, factoring, quadratics. Roughly the algebra of GCSE Higher. Chapters Chapter 5 to Chapter 8, Chapter 11.
- "He’s doing Geometry"
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A full year of Euclidean geometry with formal proof, right-triangle trigonometry, circles and coordinate geometry. Roughly the geometry of GCSE Higher, plus far more proof. Chapters Chapter 9, Chapter 10, Chapter 22.
- "He’s doing Algebra II"
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The big pre-calculus algebra course: polynomials, rational functions, complex numbers, exponentials and logarithms, sequences and series, often trigonometric functions. In UK terms, the top end of GCSE Higher plus a good part of the pure algebra in the first year of A-level. A 15-year-old in Algebra II is a year or two ahead of the standard track. Chapters Chapter 7, Chapter 11, Chapter 12, Chapter 15, Chapter 14.
- "She’s in Honors Algebra II"
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The same territory, faster and with harder problems; sometimes includes topics normally saved for Precalculus.
- "He’s doing Integrated Math II"
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A mixed course: typically quadratics, more functions, similarity and right-triangle trigonometry, probability. Roughly the second half of Algebra I combined with much of Geometry.
- "She’s doing Precalc"
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Functions in depth, full trigonometry, polar form, vectors, an introduction to limits. Roughly A-level pure mathematics without the calculus. Chapters Chapter 11, Chapter 13, Chapter 14, Chapter 16.
- "He’s in AP Calc AB" / "She’s taking BC"
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University-level single-variable calculus, ending in a national exam in May. AB is roughly the calculus of A-level Mathematics; BC adds series and polar and parametric calculus, taking it a little beyond. Chapters Chapter 16 to Chapter 19.
- "She got a 5 in BC"
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The top score on the AP Calculus BC exam. Many universities will grant credit for one or two semesters of calculus.
- "He’s doing AP Stats"
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Introductory statistics: data, sampling, probability, the normal distribution, confidence intervals and hypothesis tests. Roughly the statistics of A-level Mathematics. Chapters Chapter 29 to Chapter 31.
- "She’s taking Multivariable at the community college"
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She has finished AP Calculus BC early and is taking university-level calculus through dual enrollment. Chapter Chapter 20, and beyond this book.
- "He got a 740 on the SAT Math"
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A strong score (out of 800), reflecting secure command of Algebra I, Geometry and Algebra II material. It says nothing about calculus.
D.11. University mathematics for computer science
US computer-science degrees almost all require a common core of mathematics, usually taken in the first two years. (UK computer-science degrees cover much the same ground in first-year modules with names like "Mathematics for Computer Science" and "Discrete Mathematics".)
| Course | Typical content | Chapters |
|---|---|---|
Calculus I and II |
Limits, derivatives, integrals, techniques of integration, sequences and series (AP Calculus BC with a 4 or 5 often exempts students from both) |
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Discrete Mathematics (one or two semesters) |
Logic, sets, functions and relations, proof techniques and induction, recurrences, counting, number theory, graphs and trees. The most important mathematics course for most programmers |
Chapter 21, Chapter 22, Chapter 23, Chapter 24, Chapter 25, Chapter 14 |
Linear Algebra |
Systems of equations, matrices, vector spaces, determinants, eigenvalues, orthogonality, least squares, the SVD. Essential for graphics, machine learning and scientific computing |
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Probability and Statistics |
Probability, random variables, distributions, expectation, the central limit theorem, estimation and testing; often calculus-based |
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Multivariable Calculus (often optional for CS, required for machine learning, graphics and robotics specialisms) |
Partial derivatives, gradients, optimisation, multiple integrals |
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Numerical Methods / Scientific Computing (elective) |
Floating point, root finding, numerical integration and differential equations, numerical linear algebra |
Algorithms and theory-of-computation courses then draw on all of this, especially discrete mathematics, asymptotic notation and probability. Machine-learning courses lean hardest on linear algebra, multivariable calculus, probability and optimisation, which is exactly the combination brought together in Chapter 33.