I have been volunteering to tutor secondary-school students for their matriculation exams. This led me to write learning materials and other documents for students and teachers. This page is divided into three parts:

Learning Materials

Two-dimensional diagrams for motion and work problems

Motion and work problems are common in high-school mathematics. (Car 1 starts from city A at 10:00 while car 2 starts from city B at 11:00 ... Painter 1 starts work at 10:00 while painter 2 starts work at 11:00 ...). One typically solves motion problems by drawing the routes traveled. Both motion and work problems are also solved using tables.

This document shows how 2D diagrams of distance or work vs. time facilitate solving such problems. The diagrams are easy to draw and need not be to scale so they can be used when solving problems on exams. 

The challenge of series

I find problems on series to be relatively easy, but some students find them difficult. This document analyzes the problems on series from 13 bagrut exams (806 questionnaire) showing techniques that can be used and pitfalls to watch out for.

Mistakes in solving mathematics problems

Textbooks in mathematics are "cleaned up" and do not show all the mistakes that were made when solving problems. This document presents my solutions to some problems from the 2014 and 2015 bagrut exams (806 questionnaire) including mistakes made and lessons to be learned from the mistakes.

Visualization of theorems of Euclidean geometry

This document presents some of the more difficult theorems of Euclidean geometry needed by high-school students of mathematics. The theorems are displayed in a purely visual manner using color and other notation. The idea is that visual memory and recall of geometrical theorems might prove more efficient than verbal memory.

How to do trigonometry without memorizing (almost) anything

Trigonometry facilitates geometric reasoning using algebraic computation. To the student, trigonometry can appear as a large set of obscure formulas to be memorized. This document shows that trigonometric identities can be obtained by geometric reasoning with little memorization. Although these formulas are easy to memorize, it is useful to see how they can be proved using only geometric facts.



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.

Version 1.4.1, 29 August 2017:

How to (almost) square a circle

Given a circle, it is impossible to construct a square with the same area because the number pi is transcendental. There are rational numbers which are approximations of pi, in particular, 355/133=3.14159292. This note presents Ramanujan's construction of this number. The presentation is incremental and exercises ask the reader to perform the computations.

How to trisect an angle (if you are willing to cheat)

It is well know that it is impossible to trisect an arbitrary angle with a straighedge and compass. However, if you are willing to "cheat" and use other tools it can be done. This document shows how to trisect an angle using a simple tool, Archimedes neusis, and a more complex construct using Hippias's quadratrix. The quadratrix can also be used to square the circle.

 x+5  = 5 - x2

Solve for x. This is not an easy problem, but after (unsuccessfully) trying a few "clever" methods, I found a simple method that works.

Langford's problem

In the following arrangement of colored blocks:

There is one block between the red blocks, two blocks between the blue blocks and three blocks between the green blocks. Expressed in numbers, the bag of numbers {1,1,2,2,3,3} can be arranged in a sequence 312132 such that between the two occurrences of i there are i blocks. Langford's problem asks if this is always possible for {1,1,...,n,n}. Donald Knuth showed that solutions to Langford's problems can be easily found using SAT solvers, such as LearnSAT that I developed.

PDF (English).
PDF (Hebrew).
Zip archive with PDF and LaTeX.

Post-secondary Materials

Three-dimensional rotations

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

Version 2.0, 6 March 2017: