Mathematics

Bagrut exams (in Hebrew)

Solutions to all the questions on the Israeli Bagrut examinations in mathematics (806 / 581) from years ayin-dalet to ayin-het.

A separate document contains a rewritten and expanded version of the solutions to the problems in probability.

  • Download from GitHub.
  • The PDF can also be downloaded from Ramzor.

Mosteller's challenging problems in probability

This is a "reworking" of Fifty Challenging Problems in Probability with Solutions by Fredrick Mosteller. The reworking involves improving the accessibility of the problems for students and teachers by simplifying the problems, dividing them into subproblems, providing hints and including additional detail, calculations and diagrams. It is available in both English and Hebrew.

Markov chain simulations

Simulations written in Python 3 of problems in Markov chains: gambler's ruin, random walk (1D, 2D, 3D), Ehrenfest model, two-state process. The results of the simulation are output together with the theoretical probabilities; in addition, graphs and histograms are provided.

A functional approach to teaching trigonometry

Trigonometric functions are usually taught to secondary-school students as ratios of the sides of a right triangle with degrees as the units of measure. This can cause problems when students encounter trigonometric functions in calculus, where the functions are defined for all real numbers, and where the units of measure must be radians in order to define derivatives. This document is a guide for teachers that examines teaching trigonometry initially as functions defining by "winding" a thread around the unit circle.

The Hebrew document was written by Avital Elbaum-Cohen, at the Department of Science Teaching, Weizmann Institute of Science. The English document is my translation and adaptation of her work.

Geometry without words

An A2 sized poster with 15 theorems of secondary-school geometry displayed visually in color, but without words. It should be easier for students to see a color diagram than to understand textual theorems like "The intersection of the medians of a triangle divides the medians in the ration 2:1".

The many guises of induction

Induction is often presented as a mechanical procedure for proving properties of sequences, but it widely used in other areas of mathematics (geometry, trigonometry, logic) and in computer science (data structures, automata, formal languages). This document tries to show that induction is a uniform concept although it appears in many guises. (Some of the examples and exercises require a background in computer science and discrete mathematics.)

Version 1.6.3: 16 April 2021:

Minesweeper is NP-Complete

Richard Kaye showed that a puzzle based on the minesweeper game is NP-complete. This document presents Kaye's result with detailed explanations of the construction.

Three-dimensional rotations

This is a tutorial on Euler angles and quaternions for describing rotations in three dimensions.

Version 2.0.1, 19 July 2018:

TikZ examples

Well-documented examples of TikZ diagrams for Euclidean geometry: (1) the intersection of the perpendicular bisectors of a triangle is the center of the circumscribed circle; (2) the intersection of the medians of a triangle divide the medians in the ration 2:1; (3) Ptolemy's theorem relating the lengths of the diagonals and the lengths of the sides of a quadrilateral that is circumscribed by a circle; (4) Ramanujan's construction 355/113, an excellent approximation to pi.