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maths

117 topics across 7 chapters

Chapter 1

Mathematical foundations

Mathematical notation & sets

3 subtopics

Practice: translate between set-builder notation and roster notation (20 problems)

Practice: interval notation ↔ inequality notation (20 problems)

Memorize and use key symbols (∀, ∃, ⇒, ⇔, ∈, ⊆) with 10 self-made examples

Logic and proof techniques

4 subtopics

Truth tables: evaluate compound statements (AND/OR/NOT/⇒/⇔) (15 problems)

Practice: negating statements with quantifiers (∀/∃) (15 problems)

Prove/disprove: direct proof vs contrapositive vs contradiction (6 short proofs)

Write 5 clean proofs using definitions (even/odd, divisible, prime, injective/surjective)

Functions, graphs, and modeling

3 subtopics

Given a function, find domain/range and key features from its formula (15 problems)

Graph transformations drill: shifts, stretches, reflections (20 quick sketches)

Modeling mini-project: choose a real dataset and fit a simple functional model

Problem-solving & mathematical communication

2 subtopics

Polya-style routine: understand → plan → execute → check (apply to 10 problems)

Write full solutions with clear steps and final checks (5 problems, self-review)

Chapter 2

Numbers & arithmetic

Integers, fractions, and decimals

3 subtopics

Compute with fractions: add/subtract/multiply/divide (30 mixed problems)

Order of operations + negative numbers (25 problems)

Estimation & rounding: check reasonableness (15 problems + explain each estimate)

Ratios, rates, percent, and units

2 subtopics

Unit conversions and dimensional analysis (20 problems)

Percent change, interest, and rate problems (20 problems)

Exponents, radicals, and logarithms

3 subtopics

Exponent rules drill (product/quotient/power/negative exponents) (25 problems)

Simplify radicals and rationalize denominators (20 problems)

Logarithms: change of base, solve simple log/exponential equations (20 problems)

Modular arithmetic basics

2 subtopics

Compute remainders and use congruences (a ≡ b mod n) (25 problems)

Modular arithmetic applications: divisibility tests + clock arithmetic (10 problems)

Chapter 3

Algebra

↗ Functions, graphs, and modeling (see Chapter 1)

Algebraic expressions & manipulation

2 subtopics

Simplify algebraic expressions (factoring out GCF, combining like terms) (25 problems)

Factoring toolkit: trinomials, difference of squares, grouping (25 problems)

Linear equations & systems

2 subtopics

Solve linear equations with parameters and check solutions (20 problems)

Solve 2×2 and 3×3 systems (substitution/elimination) (20 problems)

Quadratic & polynomial equations

2 subtopics

Quadratics: solve by factoring/completing square/quadratic formula (25 problems)

Polynomials: remainder theorem + basic synthetic division (15 problems)

Inequalities & absolute value

2 subtopics

Solve compound inequalities and represent solutions on a number line (20 problems)

Absolute value equations/inequalities (20 problems)

Analytic / coordinate geometry (lines, circles, conics)

2 subtopics

Find equations of lines (slope/intercept/point-slope) (20 problems)

Circles & conics: standard forms and graphing from equations (15 problems)

Chapter 4

Geometry & trigonometry

Euclidean geometry fundamentals

2 subtopics

Angle facts and triangle basics (sum of angles, exterior angle) (20 problems)

Area/volume formulas: derive and apply (15 problems + 3 derivations)

Similarity, congruence, and transformations

2 subtopics

Prove triangle congruence (SSS/SAS/ASA/AAS/HL) on 6 diagrams

Similarity and scale factors: solve for unknown lengths (15 problems)

↗ Analytic / coordinate geometry (lines, circles, conics) (see Chapter 3)

Trigonometry fundamentals

3 subtopics

Unit circle values: memorize key angles and signs (quiz yourself daily for 1 week)

Solve right-triangle trig problems (SOH-CAH-TOA) (20 problems)

Trig identities: verify and simplify (Pythagorean, angle-sum) (15 problems)

Vectors and basic analytic geometry (dot product, projections)

2 subtopics

Compute dot products, angles, and projections (15 problems)

Vector geometry practice: distances, midpoints, and basic vector equations (15 problems)

Chapter 5

Calculus

↗ Functions, graphs, and modeling (see Chapter 1)

Limits & continuity

2 subtopics

Compute limits algebraically (factoring/rationalizing) (20 problems)

Continuity: identify discontinuities and classify them (15 problems)

Differential calculus

3 subtopics

Derivative rules drill (power/product/quotient/chain) (25 problems)

Implicit differentiation and related rates (15 problems)

Optimization & graphing with derivatives (critical points/concavity) (15 problems)

Integral calculus

3 subtopics

Antiderivatives and basic integration rules (20 problems)

Definite integrals + Fundamental Theorem of Calculus (15 problems)

Applications: area between curves and volume (disk/washer) (12 problems)

Sequences & series

2 subtopics

Sequences: compute limits and recognize geometric/arithmetic (15 problems)

Series: convergence basics (geometric, p-series, divergence test) (15 problems)

Multivariable calculus (partial derivatives, gradients, double integrals)

2 subtopics

Compute partial derivatives and gradients (15 problems)

Double integrals on simple regions (rectangles/triangles) (12 problems)

Differential equations (intro)

2 subtopics

Solve separable first-order differential equations (12 problems)

Modeling with differential equations: exponential growth/decay (6 problems + interpretation)

Chapter 6

Probability & statistics

Counting & combinatorics basics

2 subtopics

Counting rules: addition/multiplication principles (15 problems)

Permutations vs combinations (nPr, nCr) (20 problems)

Probability foundations

2 subtopics

Compute probabilities with Venn diagrams and conditional probability (20 problems)

Independence vs mutual exclusivity: classify and compute (15 problems)

Random variables & distributions

2 subtopics

Expected value and variance computations (15 problems)

Work common distributions: Bernoulli/binomial/normal (15 problems)

Descriptive statistics & data visualization

2 subtopics

Summarize datasets: mean/median/IQR/std dev (10 datasets)

Make and interpret histograms/boxplots/scatterplots (10 plots)

Statistical inference (confidence intervals & hypothesis tests)

2 subtopics

Confidence intervals: compute and interpret (10 problems)

Hypothesis tests (z/t): set up null/alt, p-value, conclusion (10 problems)

Regression & correlation (intro)

2 subtopics

Compute correlation and interpret strength/direction (10 problems)

Fit and interpret a simple linear regression line (10 problems)

Chapter 7

Discrete mathematics & algorithms

↗ Logic and proof techniques (see Chapter 1)

↗ Counting & combinatorics basics (see Chapter 6)

↗ Modular arithmetic basics (see Chapter 2)

Sets, relations, and functions (discrete viewpoint)

2 subtopics

Relations: test reflexive/symmetric/transitive; equivalence classes (15 problems)

Partial orders and Hasse diagrams (10 problems)

100

Graph theory

3 subtopics

101

Graph basics: degrees, paths, cycles, connectedness (15 problems)

102

Trees: spanning trees and basic properties (10 problems)

103

Graph algorithms practice: BFS/DFS by hand on 6 graphs

104

Algorithms & complexity basics

2 subtopics

105

Big-O intuition: compare growth rates and simplify expressions (15 problems)

106

Design 3 algorithms (search/sort/greedy) and analyze time complexity informally

107

Recurrences & induction practice

2 subtopics

108

Mathematical induction: prove 6 statements (sums, inequalities, divisibility)

109

Solve linear recurrences (simple) and verify solutions (10 problems)