Unit 1: Arithmetic and Number Sense
Computation and estimation
Order of operations and clever simplification
Fractions, decimals, and percents
Estimation and approximation

Exponents and radicals
Laws of exponents
Simplifying radicals and rationalizing denominators
Scientific notation

Word problems
Percent change, mixtures, and interest (simple)
Averages and weighted averages
Divisibility and factors
Divisibility rules
Prime factorization
Counting divisors and sum of divisors

GCD and LCM
Euclidean algorithm
Relationship between GCD, LCM, and the product

Modular arithmetic
Remainders and congruences
Units digits and cycles
Chinese Remainder Theorem (basic)

Number bases
Converting between bases
Arithmetic in other bases

Diophantine problems
Linear Diophantine equations
Perfect squares and cubes
Digit problems
Linear equations and inequalities
Solving and graphing
Systems of equationsAbsolute value equations and inequalities

Quadratics
Factoring and completing the square
Quadratic formula and discriminant
Vieta’s formulas (sum and product of roots)
Vertex form and optimization

Polynomials
Operations and factoring techniques
Remainder and Factor Theorems
Rational Root Theorem
Vieta’s formulas for higher degrees

Functions
Function notation and composition
Domain, range, and inversesGraph transformations
Functional equations (basic)

Exponential and logarithmic
Exponential growth and decay
Logarithm properties and change of base
Solving exponential and log equations

Sequences and series
Arithmetic and geometric sequences
Finite and infinite geometric series
Telescoping sums
Recursive sequences

Complex numbers (basic)
Operations and powers of i
Factoring tricks
Difference of squares, sum/difference of cubes
Simon’s Favorite Factoring Trick

Symmetric expressions
Manipulating sums and products

Inequalities
AM-GM (basic)
Trivial inequality (squares are nonnegative)

Word-problem modeling
Rate, work, distance, and age problems
Angles and lines
Parallel lines and transversals
Triangle and polygon angle sums

Triangles
Pythagorean Theorem and triples
Special right triangles
Similar and congruent triangles
Area formulas, including Heron’s formula
Triangle inequality

Special triangle lines
Medians, angle bisectors, altitudes
Angle Bisector Theorem
Centroid, incenter, circumcenter

Quadrilaterals and polygons
Properties of parallelograms, trapezoids, kites
Regular polygons and interior angles

Circles
Chords, tangents, secants
Inscribed and central angles
Power of a Point
Arc length and sector area
Cyclic quadrilaterals

Trigonometry
Sine, cosine, tangent in right triangles
Law of Sines and Law of Cosines (occasional)
Area as (1/2)ab sin C

Geometric strategies
Area ratios and shaded regions
Dissection and auxiliary lines
Mass points (occasional)
Lines
Slope, intercepts, parallel and perpendicular
Distance and midpoint formulas

Conics (basic)
Equation of a circle
Parabolas as graphs of quadratics

Areas on the plane
Shoelace formula
Transformations and reflections

Coordinate bashing as a solution strategy
Surface area and volume
Prisms, cylinders, pyramids, cones, spheres

Cross sections and slicing

3D visualization
Cubes, nets, and diagonals
Distances between points in space
Counting principles
Multiplication and addition principles
Casework and complementary counting
Overcounting and dividing out symmetries

Permutations and combinations
Arrangements with repeated items
Circular permutations
Stars and bars
Pascal’s triangle and the Binomial Theorem

Inclusion-exclusion (two and three sets)

Probability
Independent and dependent events
Conditional probability
Geometric probability
Expected value

Recursion and states (basic)

Pigeonhole Principle (occasional)
Statistics
Mean, median, mode, range
Standard deviation (conceptual)
Effect of changing data

Logic and reasoning
Truth-teller and liar problems
Games and invariants
Working backward
Draw and label a clear diagram
Try small cases and look for patterns
Use symmetry and substitution
Work backward from the answer choices
Estimate to eliminate choices
Use complementary counting
Set up coordinates for geometry
Check units and edge cases