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 several parts:
- Bagrut exams (in Hebrew only): solutions to the Israeli matriculation exams (questionnaire 806--years 2014-2108--for the 11th grade). The documents contain not just the solutions, but also proposals for new methods, guidelines for solving the problems and potential mistakes to avoid.
- Standalone documents: Learning materials on geometry and trigonometry that have been integrated into the bagrut exams document, but also available in English.
- 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.
- Instructions for formatting the documents.
Bagrut Exams
There will be three volumes, one for each "chapter" of the bagrut exam. Currently available are the first volume with questions 1 motion and work, 2 series, 3 probability and question 4 geometry.
Version 1.0.0 (1 January 2019):
Standalone Documents
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 - x^{2}
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 a^{2} = b^{2} + c^{2}.
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:
Instructions for formatting the documents
The LaTeX source files for these documents can be found on GitHub at:
https://github.com/motib/mathematics
The formatting was done using MikTeX and TeXWorks:
The Cumulus Hebrew fonts were used:
http://www.ma.huji.ac.il/~sameti/tex/culmusmiktex.html
Notes:
- TeXWorks is included in the MikTeX distribution.
- When writing Hebrew LaTeX in TeXWorks, place mathematics ($$, \[\], etc.) on separate lines to avoid difficulties with the cursor movement.