Mathematics Fluency

The number line, integers, and absolute value — the foundation of all arithmetic

Addition as forward movement and subtraction as backward movement on the number line

Multiplication as repeated addition and area, division as equal partitioning with remainders

Fractions as parts of wholes, equivalent fractions, and adding fractions with common denominators

Decimals as fraction shorthand, percentages as fractions of 100, and converting between all three

PEMDAS rules for evaluating expressions — why 2 + 3 x 4 is 14, not 20

Factors, prime numbers, prime factorization, and the greatest common divisor

Algebraic variables as placeholders for unknown values, and how to build and evaluate expressions

Isolate the variable to solve equations of the form ax + b = c using inverse operations

Solving and graphing inequalities, compound inequalities, and interval notation

Solving two equations with two unknowns using substitution and elimination methods

Polynomial structure, degree, leading coefficients, and arithmetic operations on polynomials

Breaking polynomials into factors using common factors, grouping, and special patterns

Solving ax^2 + bx + c = 0 using factoring, completing the square, and the quadratic formula

Functions as input-output machines, domain, range, and function notation

Plotting functions on the coordinate plane, understanding slope-intercept form, and visualizing parabolas

The fundamental building blocks of geometry: points, lines, rays, segments, and angle measurement

Triangle types, the Pythagorean theorem, and the triangle inequality

Circles, circumference, area, and arc length -- the geometry of curves

General polygons, computing perimeter and area, and the shoelace formula

The coordinate plane, distance formula, and midpoint -- connecting algebra to geometry

Translations, rotations, and reflections -- moving shapes on the coordinate plane

Computing volume and surface area for boxes, cylinders, spheres, and other 3D shapes

The foundation of logical reasoning: propositions, connectives, and negation

Systematically evaluating every possible truth assignment for logical expressions

Tautologies, contradictions, and the rules that let you simplify boolean expressions

Extending propositions with variables, universal quantifiers, and existential quantifiers

Building logical arguments step by step from assumptions to conclusions

Proving statements by showing their negation leads to an impossibility

Proving statements about all natural numbers using base cases and inductive steps

Combining functions by feeding the output of one into the input of another

Understanding polynomial functions, their roots, and end behavior

Functions formed by dividing polynomials, including asymptotes and discontinuities

Functions where the variable is in the exponent, modeling growth and decay

Logarithms as inverse exponentials with change-of-base and key properties

Arithmetic and geometric sequences, their formulas, and partial sums

Numbers with real and imaginary parts, enabling solutions to previously unsolvable equations

Understanding angle measurement in degrees and radians, and converting between them

The three fundamental trigonometric ratios and how to compute them from triangle sides

Using the unit circle to understand sine and cosine for any angle, including reference angles and quadrants

Key trig identities including the Pythagorean identity and double-angle formulas, and verifying them with code

Using arcsin, arccos, and arctan to find angles from trig ratios, including domain restrictions

The rigorous epsilon-delta definition of a limit, one-sided limits, and numerical approximation techniques

What it means for a function to be continuous, types of discontinuities, and the Intermediate Value Theorem

The derivative as the limit of the difference quotient, instantaneous rate of change, and the tangent line

The power rule, product rule, and quotient rule for efficiently computing derivatives

Differentiating composite functions by multiplying the derivatives of the outer and inner functions

Differentiating equations where y is not explicitly isolated, using the chain rule on both sides

Using implicit differentiation with respect to time to connect rates of change of related quantities

Finding maximum and minimum values using critical points and the second derivative test

Reversing differentiation to find antiderivatives, indefinite integrals, and the constant of integration

Approximating the area under a curve using rectangles with left, right, and midpoint methods

The definite integral as signed area, integration bounds, and the trapezoidal rule for numerical approximation

The theorem that connects differentiation and integration, showing they are inverse operations

The integration technique that reverses the chain rule by substituting a new variable

Break complex integrals into simpler pieces using the product rule in reverse

Decompose rational expressions into simpler fractions to unlock integration of rational functions

Evaluate integrals with infinite limits or unbounded integrands and determine whether they converge

Understand sequences as ordered lists of numbers and determine when they approach a finite limit

Determine whether infinite series converge using the ratio test, comparison test, and other key techniques

Represent functions as infinite polynomials and determine where those representations are valid

Approximate any smooth function with polynomials built from its derivatives at a single point

Vectors as the fundamental building blocks of linear algebra, and the spaces they live in

Matrices as rectangular arrays of numbers, and the operations that make them the workhorse of linear algebra

Solving systems of linear equations systematically using row reduction and augmented matrices

The determinant as a single number that reveals whether a matrix is invertible and how it scales space

Functions that preserve vector addition and scalar multiplication, and their matrix representations

Special vectors that a matrix only scales, revealing the fundamental behavior of linear transformations

Perpendicular vectors, projections, and the Gram-Schmidt process for building orthogonal bases

Differentiate functions of several variables one variable at a time

Combine partial derivatives into the gradient vector and measure rates of change in any direction

Differentiate compositions of multivariable functions using the chain rule and Jacobian matrices

Integrate functions of two variables over planar regions to find volumes, masses, and averages

Integrate functions of three variables over solid regions to compute volumes, masses, and averages in 3D

Assign a vector to every point in space, then measure divergence and curl to understand flow and rotation

Integrate scalar and vector quantities along curves and over surfaces to compute work, flux, and area

Counting principles, permutations, and combinations — the foundation of probability

Sample spaces, events, and the axioms that make probability rigorous

How new evidence updates beliefs — the core of Bayesian reasoning

Assigning numbers to outcomes — expected value, variance, and the language of uncertainty

Normal, Binomial, and Poisson — the distributions that model the real world

Why averages are Normal — the most important theorem in statistics

Quantifying uncertainty — how confident can we be in our estimates?

Making decisions from data — null hypotheses, p-values, and significance

Finding the line of best fit — predicting one variable from another

Ordinary differential equations that relate a function to its first derivative, and how to solve them numerically with Euler method

Separable ODEs where variables can be split to opposite sides and integrated independently

Second-order linear ODEs with constant coefficients, the characteristic equation, and the three types of solutions

Coupled systems of first-order ODEs, phase portraits, and simulating multi-variable dynamics

Laplace transforms convert differential equations into algebraic equations, making complex problems solvable with algebra

The addition and multiplication principles — the two foundational rules for counting outcomes in combinatorics

Vertices, edges, and adjacency — the mathematical structures behind networks, maps, and algorithms

Define sequences by their previous terms — from Fibonacci to algorithm analysis

Divisibility, prime factorization, and the Euclidean algorithm — the math behind cryptography and hashing

Clock arithmetic, congruences, and modular inverses — the math of cryptography and computer arithmetic

The axiomatic foundation of the real numbers — completeness, the Archimedean property, and why the rationals are not enough

The formal epsilon-N definition of sequence convergence, Cauchy sequences, and why completeness makes it all work

The rigorous epsilon-delta definition of continuity and uniform continuity — making the intuitive notion of no jumps mathematically precise

The rigorous limit definition of the derivative, Rolle theorem, and the mean value theorem with its proof and consequences

The rigorous construction of the Riemann integral via upper and lower Darboux sums, integrability criteria, and the fundamental theorem of calculus

The group — the most fundamental algebraic structure — and the four axioms that define it

Subgroups live inside groups, and cosets partition groups into equal-sized pieces — leading to Lagrange theorem

Structure-preserving maps between groups — the right way to compare algebraic structures