Math

“Early Math” refers to mathematical concepts and skills traditionally taught before Algebra. In some cases, “Pre-Algebra” may refer to the entirety of Early Math, while in other cases schools will differentiate between Early Math (pre-symbolic reasoning) and Pre-Algebra (arithmetic reasoning).

- Counting & Measurement
- Addition
- Subtraction
- Place Value
- Multiplication
- Multiplication Tables
- Basic Division
- Long Division
- Fractions
- Percentages
- Negative Numbers
- Basic Probability

Algebra is the heart of mathematical study. Put most concisely, Algebra is the mathematical framework that allows us to deduce unknown information given defined relationships with known information. Throughout their study of Algebra, students learn to model the real world on the page using the relationship between a “dependent” function f(x) and its “independent variable” x.

Almost all our day-to-day encounters with numbers, whether it be finance, cooking, planning, or anything else, implicitly demand a capability of algebraic deduction and reasoning whether we realize it or not. When students learn Algebra, therefore, they gain much more than the ability to move symbols around on the page: they learn how to think quickly, effectively, and rationally on their feet in all kinds of situations.

Most students begin studying Algebra in Middle School, often 6th, 7th, or 8th grade. Though curricula vary from school to school, many schools offer two courses entitled “Algebra I” and Algebra II” (or the equivalent). A comfortable grasp of Algebra is essential for all higher study of mathematics, including Pre-Calculus, Calculus, and beyond.

Click below to see topics traditionally covered in each course. Keep in mind that schools may cover material in different ways.

- Algebra I/A

- Linear Functions and Equations
- Systems of Linear Equations
- Inequalities
- Quadratic Functions
- Exponent Properties
- Exponential Equations

- Algebra II/B

- Absolute Value
- Square root Functions
- Cubic Functions
- Cube root Functions
- Polynomials
- Rational Functions
- Logarithmic Functions
- Data Analysis

Geometry is the study of spatial relationships. Trigonometry is a subset of Geometry that specifically examines the relationships between angles and side lengths in triangles. Topics in Geometry and Trigonometry concern the measurement and definitions of angles, lines, and shapes in any given number of dimensions. Concurrent with algebraic thinking, the practice of Geometry allows students to determine unknown information about a spatial layout based on other known properties of that layout.

In the “real” world, Geometry is most immediately relevant for any problem involving design and construction whether digital or physical. Architects and artists alike use geometric reasoning to communicate and realize their mind’s eye vision for the world.

Most students begin studying Geometry in Middle School or High School (8th or 9th grade) and complete two courses related to the subject. The progression of Geometric and Trigonometric study tends to vary more widely from school to school than other subjects: for instance, some schools have a single Geometry course and possibility a separate Trigonometry course, other schools may have two sequential Geometry courses that include Trigonometry in the curriculum, while others still may have a Geometry course or sequence and include Trigonometry in the Pre-Calculus curriculum. Like Algebra, Geometry and Trigonometry provide an essential foundation for all higher study of mathematics and physical sciences.

Given the variance of course progression from school to school, it is difficult to universally define what constitutes a Geometry course. By the end of any Geometry/Trigonometry sequence, however, students are likely to have encountered the following topics:

- Plane Geometry (Points, Lines, Planes, Angles)
- Area
- Parallel Lines & Transversals
- Similarity & Congruence
- Right Triangles (Trigonometry)
- Pythagorean Theorem
- 45-45-90 Triangles
- 30-60-90 Triangles
- Triangles (Trigonometry)
- Law of Sines
- Law of Cosines
- Quadrilaterals
- Circle Theorems
- Proofs & Conditional Logic
- High Topics may include:
- Vectors
- Transformations (using Matrices)

In the same sense that “Pre-Algebra” is sometimes loosely defined, “Pre-Calculus” can refer to any mathematical topics that provide the basis for learning Calculus. In most High School contexts, though, a Pre-Calculus class serves as a transition from the conceptual foundations of Algebra and Geometry to a higher study of mathematics. Most Pre-Calculus courses serve to emphasize and further develop topics that will be important for learning Calculus, including Trigonometry and properties of special functions, as well as introduce new topics not covered in introductory Algebra and Geometry curricula.

Most students will take Pre-Calculus during their sophomore or junior year of High School, or as an introductory undergraduate course if they did not take it in High School. As the name implies, Pe-Calculus offers an important foundation for the study of Calculus, higher mathematics, and physical sciences.

The Pre-Calculus curriculum will likely vary from school to school depending on what is covered in previous courses, but topics that are often covered in a Pre-Calculus course include:

- Complex Numbers
- Adding/Subtracting
- Multiplying
- Dividing
- Complex Conjugates
- Polar Form of Complex Numbers
- Higher Polynomial Topics
- Dividing Polynomials
- Binomial Theorem
- Fundamental Theorem of Algebra
- Composite Functions
- Trigonometry
- Inverse Trigonometric Functions
- Sinusoidal Functions
- Radians vs. Degrees
- Angle Addition
- Trigonometric Identities
- Vectors
- Definition
- Magnitude
- Scalar Multiplication
- Vector Addition
- Vector Subtraction
- Possible Higher Topics: Dot Product, Cross Product
- Matrices
- Row Operations
- Row-Echelon Elimination
- Matrix Addition
- Matrix Multiplication
- Matrices as Transformations
- Determinants of 2x2 Matrices
- Inverse Matrices
- Series
- Arithmetic Sequences & Series
- Geometric Sequences & Series
- Conic Sections
- Equation of a Circle
- Ellipses
- Hyperbolas
- Probability and Combinatorics
- Probability
- Compound Probability
- Permutations
- Combinations
- Probability Using Combinatorics

Calculus is the mathematical study of change. How long will it take my car to go from 0 to 60? How fast will a population of bacteria grow? What is the probability that I find an electron over here or over there if I look? Calculus answers these questions and many, many more.

Developed originally by Isaac Newton in the 17th Century for use in physical analysis, Calculus has since become foundational to almost all higher mathematics and scientific study. Long story short, if you want to understand how the natural world works, you have to understand Calculus.

Students who advance to Calculus in High School will likely encounter the subject in their Senior, Junior, or sometimes Sophomore year(s). For instance, many students in the Advanced Placement (AP) program take Calculus AB and Calculus BC. Calculus BC covers the material of Calculus AB in addition to several higher-level topics. In other schools, topics may be split across two courses entitled Calculus I and Calculus II (or an equivalent).

Undergraduate students wishing to study Mathematics, Physics, Biology, Economics, and other quantitative sciences will undoubtedly have to take a Calculus sequence. Most often, this sequence consists of classes titled Calculus I, Calculus II, Calculus III, and Calculus IV, though different schools may group subjects and courses differently.

Please keep in mind that different schools may title or organize courses in different ways, and curricula courses with the same name at different schools may differ. Below, however, is the “traditional” organization of courses and subjects:

- Calculus AB (or the equivalent)

- Limits and Continuity
- Limit Notation
- Graphing Limits
- Squeeze Theorem
- Continuity/Discontinuity
- Working with Asymptotes
- Intermediate Value Theorem
- Differentiation
- Power Rule
- Product Rule
- Quotient Rule
- Chain Rule
- Trigonometric Derivatives
- Inverse Function Derivatives
- Implicit Differentiation
- Higher-Order Derivatives
- Applications of Differentiation
- Related Rates
- Straight-Line Motion
- Approximations via Linearization
- L'Hospital's Rule
- Mean Value Theorem
- First Derivative Test
- Second Derivative Test
- Optimization Problems
- Integration
- Riemann Sum Approximations
- The Fundamental Theorem of Calculus
- Definite Integrals
- Indefinite Integrals & Antiderivatives
- Integration by Substitution
- Integration by Long Division and Completing the Square
- Differential Equations
- Modeling Real-World Scenarios with Differential Equations
- Verifying Solutions
- Slope Fields
- Separation of Variables
- Initial Conditions
- Exponential Models
- Applications of Integration
- Average Value of a Function on an Interval
- Position, Velocity, and Acceleration
- Finding the Area between Curves
- Volumes with Cross Sections
- Solids/Volumes of Revolution
- The Washer Method

- Calculus BC (or the equivalent)

- Limits and Continuity
- Limit Notation
- Graphing Limits
- Squeeze Theorem
- Continuity/Discontinuity
- Working with Asymptotes
- Intermediate Value Theorem
- Differentiation
- Power Rule
- Product Rule
- Quotient Rule
- Chain Rule
- Trigonometric Derivatives
- Inverse Function Derivatives
- Implicit Differentiation
- Higher-Order Derivatives
- Applications of Differentiation
- Related Rates
- Straight-Line Motion
- Approximations via Linearization
- L'Hospital's Rule
- Mean Value Theorem
- First Derivative Test
- Second Derivative Test
- Optimization Problems
- Integration
- Riemann Sum Approximations
- The Fundamental Theorem of Calculus
- Definite Integrals
- Indefinite Integrals & Antiderivatives
- Integration by Substitution
- Integration by Long Division and Completing the Square
- Integration by Parts
- Integration by Partial Fractions
- Improper Integrals
- Differential Equations
- Modeling Real-World Scenarios with Differential Equations
- Verifying Solutions
- Slope Fields
- Separation of Variables
- Initial Conditions
- Exponential Models
- Euler's Method
- Logistic Models
- Applications of Integration
- Average Value of a Function on an Interval
- Position, Velocity, and Acceleration
- Finding the Area between Curves
- Volumes with Cross Sections
- Solids/Volumes of Revolution
- The Washer Method
- Arc Length of Smooth Curves
- Parametric Equations, Polar Coordinates, and Vector-Valued Fields
- Defining Parametric Equations
- Arc Length of Parametric Curves
- Defining Vector-Valued Functions
- Motion Problems using Vector-Valued Functions
- Polar Coordinates
- Finding the Area between Polar Curves
- Infinite Sequences and Series
- Convergent/Divergent Series
- Geometric Series
- Nth-Term Test
- Integral Test
- Harmonic Series
- P-Series
- Comparison Test
- Alternating Series Test
- Ratio Test
- Absolute vs. Conditional Convergence
- Taylor/Maclaurin Series Approximations
- Radius and Interval of Convergence
- Power Series

Calculus is the mathematical study of change. How long will it take my car to go from 0 to 60? How fast will a population of bacteria grow? What is the probability that I find an electron over here or over there if I look? Calculus answers these questions and many, many more.

Developed originally by Isaac Newton in the 17th Century for use in physical analysis, Calculus has since become foundational to almost all higher mathematics and scientific study. Long story short, if you want to understand how the natural world works, you have to understand Calculus.

Undergraduate students wishing to study Mathematics, Physics, Biology, Economics, and other quantitative sciences will undoubtedly have to take a Calculus sequence. Most often, this sequence consists of classes titled Calculus I, Calculus II, Calculus III, and Calculus IV, though different schools may group subjects and courses differently.

- Calculus I

- Limits and Continuity
- Limit Notation
- Graphing Limits
- Squeeze Theorem
- Continuity/Discontinuity
- Working with Asymptotes
- Intermediate Value Theorem
- Differentiation
- Power Rule
- Product Rule
- Quotient Rule
- Chain Rule
- Trigonometric Derivatives
- Inverse Function Derivatives
- Implicit Differentiation
- Higher-Order Derivatives
- Applications of Differentiation
- Related Rates
- Straight-Line Motion
- Approximations via Linearization
- L'Hospital's Rule
- Mean Value Theorem
- First Derivative Test
- Second Derivative Test
- Optimization Problems
- Integration
- Riemann Sum Approximations
- The Fundamental Theorem of Calculus
- Definite Integrals
- Indefinite Integrals & Antiderivatives
- Integration by Substitution
- Integration by Long Division and Completing the Square

- Calculus II

- Integration
- Riemann Sum Approximations
- The Fundamental Theorem of Calculus
- Definite Integrals
- Indefinite Integrals & Antiderivatives
- Integration by Substitution
- Integration by Long Division and Completing the Square
- Integration by Parts
- Integration by Partial Fractions
- Improper Integrals
- Differential Equations
- Modeling Real-World Scenarios with Differential Equations
- Verifying Solutions
- Slope Fields
- Separation of Variables
- Initial Conditions
- Exponential Models
- Euler's Method
- Logistic Models
- Applications of Integration
- Average Value of a Function on an Interval
- Position, Velocity, and Acceleration
- Finding the Area between Curves
- Volumes with Cross Sections
- Solids/Volumes of Revolution
- The Washer Method
- Arc Length of Smooth Curves
- Parametric Equations, Polar Coordinates, and Vector-Valued Fields
- Defining Parametric Equations
- Arc Length of Parametric Curves
- Defining Vector-Valued Functions
- Motion Problems using Vector-Valued Functions
- Polar Coordinates
- Finding the Area between Polar Curves
- Infinite Sequences and Series
- Convergent/Divergent Series
- Geometric Series
- Nth-Term Test
- Integral Test
- Harmonic Series
- P-Series
- Comparison Test
- Alternating Series Test
- Ratio Test
- Absolute vs. Conditional Convergence
- Taylor/Maclaurin Series Approximations
- Radius and Interval of Convergence
- Power Series

- Calculus III

- Vectors and Geometry of Space
- 3D Coordinate Systems
- Vectors
- Dot and Cross Products
- Equation for Lines and Planes in 3D Space
- Conic Sections and Quadric Surfaces
- Vector-Valued Functions
- Vector Functions and Space Curves
- Derivatives and Integrals of Vector Functions
- Arc Length, Curvature, Normal and Tangent Vectors
- Partial Derivation
- Limits and Continuity of Multivariable Functions
- Partial Derivatives
- Multivariable Chain Rule
- Directional Derivatives and Gradients
- Tangent Planes and Linear Approximations
- Maximum and Minimum Values/Optimatization
- Lagrange Multipliers

- Calculus IV

- Multiple Integrals
- Fubini's Theorem
- Double and Triple Integrals
- Integrating in Cylindrical and Spherical Coordinates
- Change of Variables in Multiple Integrals
- Vector Calculus
- Vector Fields
- Line Integrals
- Curl and Divergence
- Parametric Surfaces and Surface Integrals
- Green's, Stoke's and Divergence Theorems
- Contour Integration/Complex Analysis
- Fourier Analysis
- Possible Further Topics

We consider “higher math” anything that goes beyond the calculus sequence and which is usually only covered at the college level. These classes can include topics from Linear Algebra, Differential Equations, Complex Variables, and basic proof writing. These courses are commonplace in many mathematics programs as well as quantitative sciences such as physics and chemistry. They often integrate Calculus (pun intended) to a more advanced extent, and form some of the foundational mathematical tools used in a lot of upper level math and science courses.

- Linear Algebra
- Ordinary Differential Equations
- Complex Variables/Analysis
- Partial Differential Equations
- Basic Proof Writing & Logic

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