Mathematics

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 for secondary-school mathematics.
  • Enrichment using only secondary-secondary mathematics .
  • Above and beyond: documents that use mathematics or computer science more advanced than secondary-school mathematics, although some of them may be accessible to students and teachers.

The LaTeX source files for these documents can be found on GitHub at https://github.com/motib/mathematics.


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.


Enrichment

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:

Help, my compass collapsed!

In a modern compass used for geometric constructs the distance between its legs can be fixed so that it is easy to copy a line segment or circle to another position. Euclid used a collapsing compass: when lifted from the paper, the legs fold up so a fixed distance cannot be maintain. At the beginning of his Elements, Euclid showed how to copy a line segment with a collapsing compass, so that any construction using can fixed compass can be done with a collapsing compass. This document presents an incremental presentation of Euclid's construction, as well as one of the many incorrect constructions that were frequently published. It also shows why Euclid chose his construction and not a "simpler" one that constructs a parallelogram.

Construction with only a compass

The Mohr-Mascheroni theorem proves that any geometric construction with a straightedge and compass can be done with only a compass. Obviously, you won't see the lines, but a line is defined by two points, so it is sufficient to construct those points. This document is based on Section 33 of 100 Problems of Elementary Mathematics: Their History and Solution by Heinrich Dorrie (Dover, 1965), as reworked by Michael Woltermann (http://www2.washjeff.edu/users/mwoltermann/Dorrie/DorrieContents.htm). The document is in LaTeX using TikZ for the diagrams. I have modified some diagrams to construct them incrementally and added extensive explanations, in particular, proofs of several theorems used in the proof.

Construction with only a straightedge

It is not possible with a straightedge alone to perform all constructions that can be done with a straightedge and compass. Poncelet conjectured and Steiner proved that a straightedge alone is sufficient provdied that  some circle exists somewhere in the plane. This document is based on Section 34 of 100 Problems of Elementary Mathematics: Their History and Solution by Heinrich Dorrie (Dover, 1965), as reworked by Michael Woltermann (http://www2.washjeff.edu/users/mwoltermann/Dorrie/DorrieContents.htm). The document is in LaTeX using TikZ for the diagrams. I have modified some diagrams to construct them incrementally and added extensive explanations, in particular, proofs of several theorems used in the proof.

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.

Are triangles with the equal area and perimeter congruent?

The right triangle with sides (3,4,5) has area 6 and perimeter 12. Is there a non-congruent triangle with the same area and perimeter? This document answers the question using an advanced mathematical concept called elliptic curves, but for the specific example of the (3,4,5) triangle, the presentation uses only secondary-school geometry and trigonometry. As a bonus, an elegant proof of Heron's formula for the area of a triangle is obtained.


Above and Beyond

The first two documents require some familiarity with logic and the concept of NP-completeness; appendices give an overview of these topics to make the documents accessible to secondary students.

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.

Pythagorean triples

SAT solving is capable of solving mathematical problems beyond the reach of "normal" mathematical techniques. This document introduces SAT solving and gives an overview of the proof by Heule and Kullman that: in any division of the natural numbers into two disjoint subsets, at least one subset will contain a Pythagorean triple, that is, three numbers such that a2 = b2 + c2.

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: