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## Blog Posts (23)

• The Number of Lattice Squares*

There are many puzzles about the number squares you can draw by using the grid points ( lattice points) on a given grid. Here is an example; The correct answer is not 9 (the number of 1x1 squares). There are many other squares you can create using the given points. These hard to catch tilted squares makes these puzzles interesting! Now we have a harder puzzle to work on! What is the total number of squares that can fit into an n x n grid? *Lattice squares are the squares whose vertices are on the grid points. There are two types of lattice squares, grid ones and the tilted ones. Let’s define a "grid square" as a square whose vertices are lattice points and sides are along the axis. (vertical squares). They are easy to create and have square number areas. A "tilted square" is a square whose vertices are still lattice points, but its sides are not along the axis. Tilted squares have whole number areas. The side length of a tilted square can easily be found by using the Pythagorean Theorem. Now, let’s have a look at a 3 x 3 squares and find the total number of grid and tilted squares that can be drawn using the lattice points. The number of grid squares that can be drawn is 9 +4 +1 = 14 Now, let’s find the number of tilted squares The number of tilted squares that can be drawn is 4 + 2 = 6. Then, the total number of lattice squares is 14 + 6 = 20 by using the points of a 3 x 3 grid. One may wonder if there is a short way of finding the number of squares for an n x n square. The questions we need to answer are; The number of grid squares in a n x n square The side length of the biggest tilted square that can be drawn in an n x n square The number of tilted squares in a n x n square The total number of lattice squares in an n x n square. Any relation among the number of tilted squares and grid squares We need to investigate all the possible squares carefully and record our findings systematically to be able to find answers to these questions. Here is a Polypad file you can work on to make drawings; You may need more grids to highlight to create different squares. Good luck! ------ ***------ SOLUTION We can start solving this puzzle by remembering another one! Famous" Checkerboard Puzzle". The answer of the Checkerboard Problem gives us the number of grid squares. To be able to find the total number of squares on a checkerboard, we need to consider that the board has 2 x 2 squares, 3 x 3 squares, 4 x 4 squares and so on other than 64 unit squares. If we organize our findings in a table. We may easily see that they follow the pattern of square numbers. Number of Grid squares in a n x n square; So for an n x n grid, the number of normal grid squares is simply the sum of the square numbers. One way to express the number of grid squares in an n x n grid is; When it comes to find the number of the tilted squares, we may discover different patterns. If you need an extra help for finding the side lengths of the tilted squares, you may have a look at the Square Areas on Grid Polypad Activity. When we organize the data for the tilted squares, one particular pattern can catch your eye. The number of √2 x √2 squares also follows the pattern of square numbers and so does 2√2 x 2√2 and 3√2 x 3√2 … The other tilted squares with the side lengths √5, √10, √13 … can be tricky to count. Be aware the symmetry of the square can make a different square now! √5 x √5 Example in a 4 x 4 grid square; There are 8 of them. If we have a closer look to 4 x 4 grid square, we see that there are 20 tilted square and 30 grid squares. Now, let’s have a look at the 5x5 case; Now there are 50 tilted squares and 55 grid squares. If you repeat the same steps for a 6 x 6 grid; We see that there are 105 tilted squares. You may realize that; In a n x n grid, the total number of grid squares and tilted squares, is equal to the number of tilted squares in a (n+1)×(n+1) grid. Now, let’s try to figure out the side length of the biggest tilted square that can fit into an n x n grid. Let “c” be the side length of the tilted square in a grid. By Pythagorean theorem a^2+b^2=c^2 and we also know that a+b can be at most n units long. a+b <= n For example in a 5 x 5 grid; you may draw “a+b” can never exceed the value of n. Let’s now try to write the side lengths of the tilted squares which will be added to the list for an 7x 7 grid. Find a + b <=7 the new values will be 6 +1 , 5+2 and 4+3 Now, let’s organize our findings about the tilted squares for each n x n grid; Here you may want to double check your results by comparing the patterns you have discovered before. Try to write the new values for 7x7 One way to express the number of tilted squares in a n x n square So the total number of lattice squares in a n x n grid can be found by These expressions can also prove our previous discovery about the total number of lattice squares in a n x n grid, the number of tilted squares in a (n+1)×(n+1) grid. One of the best outcomes of working on a problem like this is the beauty of the solution! Extension: Can we derive a formula for the total number of lattice squares in an n x m rectangular grid where n>m?

• Atatürk ve Matematik

10 Kasım Atatürk'ü anlamak için sadece savaş alanındaki dehasını yada devlet yaratma ve biçimlendirme becerisini konuşmak, okumak yetmez. Onun bilime ve eğitime verdiği değeri ve ülkemizin yeni nesillerinden beklentilerini anlamak da çok önemli. Bunu yaparken onun düşüncelerini ve fikirleri oluşturan deneyimlerini ve araştırmalarını, modern Türkiye'yi kurma amacıyla hangi kaynaklardan yararlandığını bilmek ve bu kaynaklara ulaşabilmek, onu anlamak yolunda ilk adım olabilir. Atatürk'ün hayatı boyunca 4000 kitaptan fazlasını okuduğunu biliyoruz. Atatürk'ün okuduğu kitapların, 1741'inin Çankaya Köşkü, 2151'nin Anıtkabir, 102'sinin İstanbul Üniversitesi Kütüphanesi ve 3'ünün ise Samsun İl Halk Kütüphanesi'nde olduğu biliniyor. Sadi Borak tarafından yazılan kısa metinde, Atatürk'ün bu kitapları okurken aldığı notlar şu şekilde açıklanmış; Bu 10 Kasım'da, O'nun fikirlerinin temellerini oluşturan kitaplara bir göz atalım. Bu kitapları okumak, onu anlamak yolunda, başkalarının fikirlerini dinlemek yerine atabileceğimiz en somut adım olacaktır. Aşağıdaki interaktif Google sınıfını buradan indirip, linklere ve videolara ulaşabilirsiniz. 23 Nisan Yakında .. 19 Mayıs Yakında .. 29 Ekim Yakında ..

• Flextangles

Flextangles are paper models with hidden faces. They were originally created by the mathematician "Arthur Stone" in 1939 and became famous when Martin Gardner published them in December 1956 issue of The Scientific American. Although you can find many different examples and ready to use templates on the web, the best method is to create your own template by using an interactive geometry software like GeoGebra. As a class activity creating flextangles by using a software can lead to discussions about translation and reflection. Flextangles, gizli yüzleri ortaya çıkarmak için esnetilebilen kağıt modellerdir. İlk olarak 1939'da Matematikçi Arthur Stone tarafından yaratılan flextangles, Martin Gardner'ın 1956 Aralık ayında The Scientific American'da yayınladığı makalede yeralınca, ünlü hale geldi. Webde bir çok örneğini ve taslak çizimlerini bulabileceğiniz flextangles için, GeoGebra gibi herhangi gibi geometri programı kullanarak kendi tasarımlarınızı da yaratabilirsiniz. Flextangle ları bir sınıf aktivitesi olarak program yardımıyla tasarladığınızda öteleme ve yansıma konularında da pratik sağlıyor. Ready to use Templates / Kullanıma Hazır Taslaklar: ------ ------ ------

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## Pages (30)

• Lessons | MATH FAN

FUN MATHFAN LESSONS NUMBERS Out of gallery NUMBER SUMS PRIMES Pi RECURRING DECIMALS BINARY SYSTEMS EXPONENTIAL GROWTH COMPLEX NUMBERS GEOMETRY Out of gallery CONSTRUCTIONS WITH CIRCLES A CLOSER LOOK TO CUBE PLATONIC SOLIDS ANTI-PRISMS SHAPES W/ CONSTANT WIDTH ARCHIMEDES I - II - III 4TH DIMENSIONAL CUBE MORE... Out of gallery Out of gallery Out of gallery Out of gallery Out of gallery INFINITY TOPOLOGY GRAPH THEORY OPTICAL ILLUSIONS STATISTICS Out of gallery PYTHAGORAS PASCAL TRIANGLE CRYPTOLOGY ALGORITHMS LEONARDO DA VINCI MATH & ART Out of gallery SPIROGRAPHS LATTICE SQUARES TESSELLATION IMPOSSIBLE SHAPES VEDIC MATHEMATICS STRING ART Out of gallery CYLINDRICAL MIRROR AMBIGIOUS SHAPES FRACTALS ORIGAMI IN SPACE X TABLE BRIDGES

• Blog | MATH FAN

Eda Aydemir The Number of Lattice Squares* There are many puzzles about the number squares you can draw by using the grid points ( lattice points) on a given grid. Here is an... Eda Aydemir Atatürk ve Matematik Bu 10 Kasımda, O'nun fikirlerinin temellerini oluşturan kitaplara bir göz atalım. Eda Aydemir FLEXTANGLES Flextangles are paper models with hidden faces. Flextangles, gizli yüzleri ortaya çıkarmak için esnetilebilen kağıt modellerdir. Eda Aydemir Net of a Sphere, Different Map Projections, Codex Atlanticus, and a Library in Italy We know that it is not possible to draw the net of a sphere like cylinders, cones, or polyhedra. So, how we can represent a 3D sphere on... Eda Aydemir Kürenin Açınımı, Farklı Haritalama Teknikleri ve Milano'da bir kütüphane; Milano'daki Ambrosiana Kütüphanesi, Matematikçiler için kutsal yerlerden bir olabilir. Eda Aydemir Create Your Own Math Clock A blank wall clock turns to a STEM activity with science, history and many different mathematical concepts like fractions, circles, angles, Eda Aydemir Çarpım Tablosunun Resmini Çizelim! Vedic Kareleri & Vedic Kurtçukları ve Spirolaterals Aşağıdaki görsellerin çarpım tablosunun kendisi olduğunu söylesem? Şimdi kendi çarpım... Eda Aydemir Vedic Squares & Vedic Worms, Spirolaterals IDENTIFYING THE PATTERNS OF MULTIPLICATION TABLE Can you draw the picture of the multiplication table? What if I tell you the images... Eda Aydemir Matematik & Sanat; İlhan Koman Sanatı ve matematiği harmanlayarak en etkileyici örneklerini yaratan ünlü heykeltıraşımız İlhan Koman’ın seneye doğumunun 100. yılını... Eda Aydemir Açık Eğitim Hareketi (Open Education Movement & Resources "OER") Eğitimcilerin, öğrencilerinin ihtiyaçlarına göre sürekli olarak hazırladıkları ve yeniledikleri eğitim materyalleri, aktif ders saatleri... Eda Aydemir Matematiğin Peşinde Ekibi Utku Aytaç, Can Ozan Oğuz Eda Aydemir MATH & SCIENCE DAYS TO CELEBRATE Celebrating birthdays of the great mathematicians and scientists as well as the national and global days related with math & science can... Eda Aydemir Profesyonel Gelişim (Professional Development ‘PD’) Funmathfan platformu online olduğundan beri en çok gelen ve en sevindirici mailer, mesleğe yeni adım atacak olan öğretmenlerin, okul... Eda Aydemir 14 Mart Uluslararası Matematik Günü Unesco'nun Kasım 2019'da, Pi günü olarak kutlanan 14 Mart gününü Uluslararası Matematik Günü ilan etmesi ile dünyanın dört bir yanındaki... Eda Aydemir SPIROGRAPH As we all know through play, kids learn different things without even realizing it! Playing with a spirograph, experimenting and trying... Eda Aydemir Adaptive Learning When Bloom published the two-sigma problem in 1984, he shared two fundamental findings. The first one is that “everyone can achieve” and... Eda Aydemir Activities Section of Mathigon Mathigon, one of the best mathematical sources of the web, also has another treasure "activities" In this section, you can find an... Eda Aydemir Virtual STEAM Museums As you all know there are great math, science and technology museums all around the world. I have listed a few of them under the STEAM... Eda Aydemir The Year Game from NCTM As a part of NCTM ’s year-long 100th birthday celebration, they have created a year game for 2020. The Game is how many expressions you... Eda Aydemir Christmas Lectures by Royal Institution Started by Michael Faraday in 1825, and now broadcast on UK national television every year, the CHRISTMAS LECTURES are the UK's flagship... Eda Aydemir Quanta Magazine I would like to introduce you "Quanta Magazine" https://www.quantamagazine.org/ Quanta Magazine is an editorially independent online... Eda Aydemir Secrets of the Surface The world premiere screening of Secrets of the Surface: The Mathematical Vision of Maryam Mirzakhani will take place on Friday, January... Eda Aydemir Math Letters to Parents ✉️ Recently, I have seen a great way of reaching out to parents on mathematical notes via a website called Big Ideas Maths. All of these...

• Online Games | MATH FAN

Online Games and Puzzles with Polypad Single Cuts Square Puzzle Star Wars Battleship LS Star Wars Battleship DS Geomagic Squares Isometric Puzzle Orange Game SUDOKU Magic Squares Hexagon Puzzle Number of Triangles Geomagic Rhombuses Square puzzle Sticky Numbers Geomagic squares 3 Pentomino Pairs HEX Mastermind Game HIP Game

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