Students in the United States begin studying mathematics at around five or six years of age, continuing through secondary school and into higher education. In elementary school, children are introduced to basic mathematics, and the theories and methods covered in math classes become increasingly complex as students age. Many schools will offer different levels of classes as students may show a greater or lesser aptitude for complex math courses.

During elementary school, students are taught basic arithmetic: addition, subtraction, multiplication and division. These concepts are elaborated on in middle school, where students will study basic algebra and concepts of variable, integers and polynomials. Many students will have completed some form of pre-algebra or even algebra 1 by the time they enter high school, although geometry is occasionally taught in eighth grade as an honors course. In high school, the general math curriculum includes algebra 1, algebra 2 and geometry in ningth and tenth grades. High school mathematics can continue with the study of algebra 3, otherwise known as trigonometry, around 11th grade. Students will complete their high school math courses senior year with either pre-calculus or calculus, although that is usually only offered at an honors level.

## Algebra

Algebra is an area of mathematics that focuses on the rules of operations and relations, and the constructions and concepts that arise from them. Subjects within algebra include terms, polynomials, equations and algebraic structures.

Algebra is a required mathematics class in all 50 states and is taught in several different stages. While stages may vary from school to school, the general stages include pre-algebra, algebra 1 and algebra 2. Pre-algebra is often taught at the middle school level, and introduces the basic concepts of polynomials and variables, thus bridging the gap between basic arithmetic and advanced algebra. Elementary algebra that is, the beginning levels of algebra introduces the concept of variables representing numbers.

Some districts introduce algebra 1 to middle school students as an honors class, though for the most part, this stage is taught at the high school level. In high school, students must complete algebra 1 and algebra 2, followed by trigonometry or pre-calculus (algebra 3). The complexity of these subjects increases as the grade level increases, but they all generally incorporate elements of powers, roots, polynomials, quadratic functions, coordinate geometry, exponential and logarithmic functions, probability, matrices and basic to advanced trigonometry.

## Geometry

Geometry is an area of mathematics concerned with questions of size, shape, relative positions of figures and the properties of space. Geometry deals with measurements, such as volume, length, angles, proofs, area, circumferences, etc. It includes algebraic forms, such as Cartesian coordinates. Geometry also overlaps somewhat with trigonometry, serving as a foundation for a more specialized area of mathematics. The study of triangular shapes is introduced in geometry, which introduces students to the basic concepts of trigonometry.

Geometry is sometimes taught as early as eighth grade as an honor’s class, though it becomes a fundamental part of the general curriculum in high school. Elementary geometry builds off the general arithmetic students learn in elementary and middle school. It is most commonly taught beginning in tenth grade. Geometry lessons are often taught in the form of queries requiring step-by-step proofs which the student must develop.

## Trigonometry

Trigonometry comprises what is sometimes known as algebra 3. Along with pre-calculus, it constitutes the later part of a student’s secondary school mathematics education. Trigonometry focuses on the study of triangles, specifically the relationships between sides and angles, as well as trigonometric functions and the motion of waves.

## Calculus

Calculus is one of the higher levels of mathematics and is only taught to secondary school students. The level of complexity in this area of mathematics is very advanced and incorporates concepts from all levels of algebra, trigonometry and pre-calculus. It focuses on limits, functions, derivatives, integrals and infinite series. Calculus requires a solid foundation in mathematics for students to grasp the various concepts.

Following the completion of algebra and trigonometry, high school students, begin studying calculus in several stages. Pre-calculus is the most common and most widely taught form of calculus. Calculus is almost always an honors-level class, if it is even taught at all, because of its highly advanced content. It is usually not taught as part of a normal curriculum before 11th or 12th grade in high school.