Caveat Academy course · Grades 10 to 12

Discrete Mathematics

A semester of the mathematics behind computing and careful reasoning: truth tables and proofs, sets, functions and induction, permutations and combinations, primes, the Euclidean algorithm and check digits, graphs, Euler paths and shortest routes, and recursion, algorithm speed and binary logic. Six modules at your own pace, with a check in every lesson and a certificate at the end.

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$19one-time · every device you sign in on
Buy · $19

Included with the Caveat Academy pass and Whole-school licenses. Or get it in the High school electives library, $119 for 66 items.

6 modules · 18 lessonsabout 13 hours$19

Six modules, eighteen lessons

Each module ends with a practice set that tracks mastery, an explain-it-back prompt, a short written response and a module exam. The final opens when every module is proficient; passing it earns a certificate with a verification code.

  1. Module 1: Logic and proof

    Build truth tables, form converses and contrapositives, negate quantified statements, and write direct proofs and proofs by contradiction.

    • Statements and truth tables
    • Conditionals and quantifiers
    • Writing proofs
    about 2.1 h

  2. Module 2: Sets, functions and induction

    Compute with sets and inclusion-exclusion, classify functions, apply the pigeonhole principle and prove formulas by induction.

    • Sets and Venn diagrams
    • Functions and the pigeonhole principle
    • Mathematical induction
    about 2.1 h

  3. Module 3: Counting

    Decide whether order matters and compute counts with the multiplication principle, permutations, combinations and Pascal's triangle.

    • The basic counting principles
    • Permutations and combinations
    • Pascal's triangle and the binomial theorem
    about 2.1 h

  4. Module 4: Number theory

    Factor integers, compute gcds with the Euclidean algorithm, and use modular arithmetic for check digits and simple ciphers.

    • Divisibility and primes
    • Greatest common divisors
    • Modular arithmetic and check digits
    about 2.1 h

  5. Module 5: Graphs and networks

    Model networks as graphs and decide when Euler paths exist, build minimum spanning trees and find shortest paths.

    • Graphs, degrees and the handshake lemma
    • Euler and Hamilton
    • Trees and shortest paths
    about 2.1 h

  6. Module 6: Recursion, algorithms and binary logic

    Solve recurrences, compare algorithms by growth rate, convert binary numbers and evaluate logic gate circuits.

    • Sequences and recurrence relations
    • Algorithms and efficiency
    • Binary numbers and logic gates
    about 2.1 h

Try it before you start

This is how every lesson works: step through a worked example at your own pace, then answer a check and get the reason right away. Nothing here is saved; the course keeps your progress.

Module 3 · Permutations and combinations

Worked example: how many poker hands

  1. A hand is 5 cards from a 52-card deck, and the order you receive them does not matter, so it is a combination.

A lesson check from each module

The checks need JavaScript turned on. The course itself works the same way.

What you get

Academics · High School$19 · included with the Academy pass and Whole-school licenses

What you finish with
A finished semester of discrete mathematics: you can build truth tables and spot a converse passed off as a proof, write direct proofs, proofs by contradiction and proofs by induction, count with permutations, combinations and inclusion-exclusion, factor numbers and find a gcd by the Euclidean algorithm, compute check digits and clock arithmetic, decide from vertex degrees whether an Euler path exists, build a minimum spanning tree, solve simple recurrences, compare algorithms by growth rate and convert to and from binary. Passing the final earns a Caveat certificate of completion.
Who it is for
Students in grades 10 to 12 who have finished Algebra 1 and want the mathematics behind computer science, puzzles and proofs, are preparing for a discrete math, computer science or math competition course, or are thinking about careers in software, data, security or engineering; homeschool families who want a complete semester course. Paper, a pencil and a basic calculator are all it needs.
Format
An online self-paced course: 6 modules and 18 lessons with worked examples you reveal step by step, 144 practice and exam items (multiple choice, numeric, fill-in and ordering), an explain-it-back prompt and a rubric-checked short response in every module, module exams with one retake, a 24-question final and a certificate with a verification code. About 13 hours in all; works on a phone or a computer.
Access
$19 one-time, yours on every device you sign in on; the course opens once you are signed in with it. Included with the Caveat Academy pass and Whole-school licenses. Also in the High school electives library ($119) and the Academics Suite pass ($149), and 1 more bundle. Progress is saved in this browser on this device, and the certificate page lets you download your record.
Where it lives
The High School space in Caveat Academics.
Support
Access, a receipt, a refund or a correction to the course: team@caveatacademic.com.

How it works

  1. Open the course and start Module 1. Each lesson takes about 25 to 35 minutes; two or three a week finishes the course in about a semester.
  2. Answer the checks as you read. Every answer shows the reason, and a miss links you back to the lesson that covers it.
  3. Finish each module with its practice set and exam. Proficient on every module opens the final; 70 percent passes it.
  4. Print or download your certificate and record from the certificate page.

Start the course

Sources and accuracy

The lessons are original writing by Caveat Academic Services. Facts were checked against these free, open references, which are also good places to read further.

  1. Oscar Levin, Discrete Mathematics: An Open Introduction, a free open textbook: discrete.openmathbooks.org
  2. MIT OpenCourseWare, 6.042J Mathematics for Computer Science, free lecture notes: ocw.mit.edu
  3. International ISBN Agency, ISBN Users' Manual, on the ISBN-13 check digit: isbn-international.org
  4. Euclid's Elements, Book VII (the Euclidean algorithm) and Book IX, Proposition 20 (infinitely many primes), in David Joyce's free online edition: mathcs.clarku.edu

Discrete Mathematics is a self-paced course from Caveat Academic Services. It follows the sequence of a typical one-semester high school discrete mathematics course but does not replace a school course and awards no credit. Historical dates are the commonly cited ones; examples such as link costs and survey counts are made up for practice and labeled as examples. Support: team@caveatacademic.com.