Teacher · PhD student · Barcelona
Àlex Reyes
Manuel Alejandro Reyes Jiménez
Maths teacher at Col·legi Maristes Sants—Les Corts and PhD student at UPC, working on the Collatz conjecture and symbolic dynamics. This is my workspace — where the classroom and research meet.
Teaching
- School
- Maristes Sants—Les Corts
- At this school since
- 2018
- Teaching since
- 2011
- Position
- Mathematics teacher (ESO, BTL & IB)
- Form tutor
- 2nd ESO
PhD
- Institution
- FME — UPC
- PhD started
- 2026
- Research line
- Symbolic dynamics — Collatz
About
Since I was a kid, I've been fascinated by problem solving. I never cared whether the solution had a practical application — I just enjoyed thinking, analysing, getting to the bottom of things. I graduated from the FME at UPC in Mathematics, and also completed an MSc in Applied Mathematics and an MSc in Teacher Training.
I spent 7 years at Escola Meritxell in Mataró as a mathematics teacher for ESO and Bachillerato — the last two as Head of Studies for Bachillerato. For the last 8 years I've been teaching at Col·legi Maristes Sants—Les Corts — the same school where I did all of my primary and secondary education.
Fifteen years in the classroom brought me back to something I had long set aside: the drive to solve real problems. One day I came across the Collatz conjecture — a problem that captures everything I love about mathematics. And thanks to that, I'm currently doing a PhD under Dr. Jaume Franch at UPC.
Right now
Working on Paper 2, focused on fibers, strong couplings and the $h_m=4$ conjecture.
Starting the 2026–2027 school year: Maths in 2nd, 3rd and 4th ESO, 1st year Bachillerato (Social Sciences), IB AI in 2nd year Bachillerato, Projects and form tutoring in 2nd ESO, and Copa Cangur coaching.
Latest updates
Teaching
IB Math AI HL4
- 5 Oct 2026 More notes for Unit 9 · Matrices in IB Math AI HL: operations, powers, determinant and inverse, systems and transformations of the plane, with the homework for 23 Sep now corrected.
- 25 Sep 2026 Unit 9 · Matrices begins in IB Math AI HL: definitions and types of matrix, the homework for 23 Sep and some matrix powers that turn into Fibonacci.
- 25 Sep 2026 New IB Math AI HL exam: U8 — Sinusoidal functions and complex numbers, with step-by-step solutions and mark scheme.
- 25 Sep 2026 Adding two sinusoids with complex numbers, with a simulator to watch the result live, and exercises 8F.1 and 8F.6 in full.
Copa Cangur9
- Jun 12, 2026 Historical archive begins: 11th Copa Cangur Final · Júnior (La Garriga, May 2025), 14 problems solved in three languages.
- Jun 12, 2026 Round 1, morning · 12th Copa Cangur · Cadet (Jan 28, 2026): 12 problems solved and explained in three languages.
- Jun 12, 2026 Round 1, morning · 12th Copa Cangur · Júnior (Jan 28, 2026): 12 problems solved and explained in three languages.
- Jun 12, 2026 Round 1, afternoon · 12th Copa Cangur · Cadet (Jan 28, 2026): 12 problems solved and explained in three languages.
- Jun 12, 2026 Round 1, afternoon · 12th Copa Cangur · Júnior (Jan 28, 2026): 12 problems solved and explained in three languages.
- Jun 12, 2026 Semifinal A · 12th Copa Cangur · Cadet (La Garriga, May 5, 2026): 10 problems solved and explained in three languages.
- Jun 12, 2026 Final · 12th Copa Cangur · Cadet (La Garriga, May 5, 2026): 14 problems solved and explained in three languages.
- Jun 12, 2026 Semifinal A · 12th Copa Cangur · Júnior (La Garriga, May 5, 2026): 10 problems solved, with the original figures.
- Jun 11, 2026 New Cangur section: final of the 12th Copa Cangur · Júnior (La Garriga, May 5, 2026), 14 problems solved one by one.
Applied Maths · CCSS25
- 8 Oct 2026 New Real numbers sections for Applied Maths CCSS (1st BTL): logarithms, their properties and their uses (Richter, decibels and the stock market, with interactive charts), plus the marked exercises.
- 25 Sep 2026 New Real numbers sections for Applied Maths CCSS (1st BTL): rationalising and the homework for 28 Sep, with folded solutions.
- 17 Sep 2026 New Real numbers sections for Applied Maths CCSS (1st BTL): intervals, powers and roots, and the marked homework.
- May 28, 2026 New Applied Maths CCSS (1st BTL) exam: Comprehensive — 3rd term (May 28, 2026), with step-by-step solutions.
- May 19, 2026 New Applied Maths CCSS (1st BTL) exam: Partial 2 · 3rd term — Probability (May 19, 2026), with step-by-step solutions.
- May 14, 2026 Applied Maths CCSS (1st BTL) notes: Binomial distribution (Unit 10) — Bernoulli, B(n,p), a worked example and 6 exercises.
- 28 Apr 2026 Probability notes (U9): Types of events — compatible/incompatible, dependent/independent, with diagrams and 4 exercises.
- 27 Apr 2026 Real functions notes (U5): Images and preimages — definitions, several examples and 4 solved exercises.
- 27 Apr 2026 Real functions notes (U5): Functions and domain — definition, 5 examples by type and 21 exercises.
- 27 Apr 2026 Inequalities notes (U4): first-degree inequalities — rules, 4 examples and the solution on the number line.
- 27 Apr 2026 Applied Maths CCSS (1st BTL) sheet: exponential and logarithmic equations and systems — 59 solved exercises, with PDF.
- 27 Apr 2026 Equations and systems notes: non-linear systems — substitution as the engine, an intro example and 8 cases.
- 27 Apr 2026 Equations and systems notes: absolute-value equations — the critical-points method and examples.
- 27 Apr 2026 Equations and systems notes: radical equations — strategy, a worked example and 10 solved exercises.
- 27 Apr 2026 Equations and systems notes: logarithmic equations — the two key identities and 11 solved exercises.
- 27 Apr 2026 Equations and systems notes: exponential equations — four types, one example per type and exercises.
- 27 Apr 2026 Equations and systems notes: review session — 17 solved equations (polynomial, biquadratic and rational).
- 27 Apr 2026 Equations and systems notes: rational equations — a five-step strategy and two solved exercises.
- 27 Apr 2026 Equations and systems notes (U3): biquadratic equations — the substitution xⁿ=z, three examples and exercise 30.
- 27 Apr 2026 Polynomials notes: GCD and LCM — the property P·Q=GCD·LCM, two examples and three exercises.
- 27 Apr 2026 Polynomials notes: Homework — five exercises (numeric value, Ruffini, factorisation) solved.
- 27 Apr 2026 Polynomials notes: Factorisation — common factor, quadratics, Ruffini and 8 worked examples.
- 27 Apr 2026 Polynomials notes (U2): monomial, polynomial, degree, numeric value, roots and factors.
- 26 Apr 2026 Full Probability unit, Applied Maths CCSS (1st BTL): sets (Morgan app), total probability, Bayes and 6 exercises.
- 26 Apr 2026 Published the full subject information for Applied Maths CCSS (1st BTL): schedule, criteria, calendar and syllabus.
3rd ESO9
- 13 Oct 2026 Homework (14 Oct) for 3rd ESO: combined operations with fractions, exercises 42 and 43.
- 13 Oct 2026 Rational numbers: fractions and operations for 3rd ESO: simplest form, equivalent fractions, addition, subtraction, multiplication and division, with the 7 October homework and the 13 October exercises worked out.
- 5 Oct 2026 Homework (5 Oct) for 3rd ESO: which set each number belongs to, true or false, and a company's profit, with solutions.
- 5 Oct 2026 Unit 1: Rational numbers for 3rd ESO. Number sets and fractions on the number line, with an interactive number line where you change the numerator and denominator.
- 25 Sep 2026 Challenge B: Renovating our first flat for 3rd ESO: the cost of painting a flat and fitting parquet and skirting board, with solutions.
- 25 Sep 2026 PDF worksheets for 3rd ESO: the problem sheets for both challenges and the formula sheet for areas and volumes.
- 23 Sep 2026 Homework for the next class for 3rd ESO: how much does it cost to paint and fill a pool shaped like a prism and half a cylinder?
- 23 Sep 2026 Homework corrected (21 Sep) for 3rd ESO: base, lateral and total area and volume of a prism, a cylinder and a pyramid, with the net of each solid.
- 23 Sep 2026 First 3rd ESO notes: the Survival laboratory, three challenges to revise 2nd ESO with the solution to every task.
4th ESO6
- Oct 4, 2026 Solutions to the 3rd ESO revision for 4th ESO, folded in each part and as a PDF.
- 30 Sep 2026 Geometry formula sheet for 4th ESO: areas and perimeters of plane figures, and AB, AL, AT and volume of solids. As HTML and PDF.
- 30 Sep 2026 3rd ESO revision for 4th ESO: seven exercises on percentages, volumes and areas to prepare for the test on 5 October.
- 30 Sep 2026 Challenges 3 and 4 for 4th ESO: fencing a football pitch with an equation, and why 0.999… equals 1, with an interactive figure.
- 29 Sep 2026 Challenge 2 for 4th ESO: a cone full of water on the roof — will it hold? Pythagoras, volume, conversions and scale, with an interactive cone.
- 23 Sep 2026 First 4th ESO notes: Challenge 1 from the maths challenges unit, a mural worked through step by step (Pythagoras, scales, areas and percentages).
2nd ESO7
- 30 Sep 2026 Fractions for 2nd ESO: the marked exercises from 25 Sep and the revision exercises for the exam, with folded solutions.
- 25 Sep 2026 New Fractions sections for 2nd ESO: multiplying and dividing, powers and roots, two sets of marked homework, the homework due 25 Sep (folded solutions) and the exam topics.
- 17 Sep 2026 New Fractions sections for 2nd ESO: equivalence and simplifying, ordering, marked homework, and adding and subtracting.
- 11 Sep 2026 First Fractions notes for 2nd ESO published for the 2026–27 year.
- 25 Oct 2025 Published exam Unit 1 — Fractions and decimals for 2nd ESO group A, with detailed solutions per question.
- 1 Oct 2025 Published exam Unit 1 — Fractions (PREVIEW) for 2nd ESO group A.
- Sep 2025 Complete Fractions unit notes published for 2nd ESO.
PhD
Publications2
- 28 May 2026 Preprint published on arXiv: “A Fibonacci theorem for Collatz trajectories via modular graph structure” (arXiv:2606.02621 [math.NT]).
- Mar 2026 First draft of the paper “A Fibonacci theorem for Collatz trajectories via modular graph structure”.
Notes & outreach3
- 22 Jun 2026 New general-audience note: “The man and the marginal note”, the origin of the Collatz conjecture and the man who jotted it down, with an interactive timeline and orbits.
- 13 Jun 2026 New general-audience note: “Fibonacci hidden in Collatz”, with interactive simulators of the mod-6 graph and the F(m+1) theorem.
- 11 Jun 2026 New interactive visualizations of the PhD research: bijections Φ/Ψ, graph G mod 6 and the binary module.
Conferences
Coming soon.
Milestones2
- Apr 2026 Started the PhD at UPC with Dr. Jaume Franch.
- Feb 2026 Progress on the formalisation of fractal modular dynamics over the Collatz conjecture.