# Mathematics

The documents are copyrighted under the CC-BY-SA license (unless otherwise noted), which permits freely downloading, copying, printing, etc. You may also modify the document provided that my name remains and that you distribute it under the same license.

## Bagrut Exams

פתרונות לכל השאלות בבחינות הבגרות במתמטיקה, שאלון 806, מהשנים תשע"ד עד תשע"ח.

.המסמך כולל גם פתרונות חלופיים, הנחיות לתהליך הפתרונות, מלכודות שיש להימנע מהם, ועוד

.נספח ב' ייצוג גרפי של משפים בגיאומטריה: קובץ נפרד להדפסה בצבע

.המסמך גם מופיע ברמזור למורה

."ניתן להוריד ולהעתיק את המסמך בחינם לפי תנאי רישיון "ייחוס-שימוש-לא-מסחרי-שיתוף זהה

## Enrichment

### 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.

### An alternate method for solving quadratic equations

Po-Shen Loh developed an alternate method for solving quadratic equations that does not require memorization of the traditional formula. This document presents Loh's method with additional examples and details of the computation.

### 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 1.6.2: 21 January 2020:

### Surprising constructions with straightedge and compass

The following interesting constructions are presented in full detail at the level of secondary school mathematics:

1. A "collapsing" compass perform any construction that a "fixed" compass can. This is theorem 2 of Euclid's Elements, but many subsequent proofs are incorrect.
2. Trisection. Although angles cannot be trisected using a straightedge and  compass, the construct can be performed using other tools.
3. It is not possible to square a circle with straightedge and compass, because pi cannot be constructed. However, there are very precise approximations that can be constructed.
4. Any construction possible with straightedge and compass can be performed with a compass only!
5. Any construction possible with straightedge and compass can be performed with a straightedge only, provided that a single circle exists somewhere in the plane.
6. Can there be non-congruent triangles with the same perimeter and area? The answer is yes! Here we show how to derive the values 156/35, 101/21, 41/15 of the edges of the triangle that have the same perimenter and area as the triangle with sides 3, 4, 5. (This results is not about construction with straightedge and compass but I included it anyway.

Version 2.1 (5 September 2020).

### The mathematics of the origami

Origami is the art of paper folding. There is a mathematical formalization of origami as an axiomatic system. Geometric constructions can be done with origami that are impossible with straightedge and compass. This document presents the mathematics of origami using only secondary-school mathematics, primarily analytic geometry. Chapters:

• Axioms
• Trisecting an angle
• Doubling a cube
• Lill's method for finding roots of cubic polynomials
• Beloch's fold and Beloch's square
• Constructing a nonagon (regular polygon with 9 sides)

I would like to thank Oriah Ben-Lulu for introducing me to this topic.

Version 4 (27 September 2020)

### Surprising geometric constructions

Errata: p. 80, line -5, delete a_3.

Here is an updated version with a corrected proof that in Axiom 6 a fold of the focus of a parabola onto the directrix is a tangent.

### Two problems in probability

This document is based upon the first two problems from Frederick Mosteller. Fifty Challenging Problems in Probability with Solutions, Dover, 1965. The problems are appropriate for secondary-school students. My solution of the first problem is different from Mosteller's, demonstrating that problems in mathematics can have multiple solutions. The second problem is interesting because the solution is counter-intuitive. Mosteller shows that the counter-intuitive solution is obvious once the problem is analyzed.

## Mathematics and Computer Science

### 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.

### 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:

### 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.

Submitted to TeXample.net.

### 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:

https://miktex.org

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.