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  • 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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  • Puzzles and Games | MATH FAN

    Fun Mathfan Shop Read. Watch. Play. Explore. Create Books Young Readers Mathflix P+ Games Toys and Gadgets 3D Models Puzzles and Games Prime Climb Explore mathematical structure in multiplication, division, and prime numbers by Math for Love Buy on Amazon ​ Q.bitz Jr Pattern identification game for younger players (1-4 players) Buy on Amazon ​ Harry Potter Trivia Game Only If you are a true HP fan Buy on Amazon ​ Mastermind A Strategy game for kids to explore the concepts of probability, deductive reasoning, and logic Buy on Amazon ​ Da Vinci Clock Model - Reproduction of a clock based on an escapement sketched by Leonardo da Vinci Buy on Amazon ​ Pattern Explorer 2 Book - diverse collections of pattern problems for students to explore, investigate, discover, and create. Buy on Amazon ​ Shot the Box Excellent tool for teaching basic addition Buy on Amazon ​ Sequence A strategy game for ages 7+ Buy on Amazon ​ Pattern Blocks A toy that every kid needs to have -Explore shapes, relations, fractions, symmetry, area measurement, and more Buy on Amazon ​ Logic Puzzles Book - 60 Clever Brain Games and Puzzles Buy on Amazon ​ Lego Chess Set A collectible for lego and chess lovers Buy on Amazon ​ Domino Set - 1000 pc Colorful Dominos Tiles for Building, Stacking, Racing, Tumbling. Buy on Amazon ​ Reversi A classical game also known as Othello - magnetic version Buy on Amazon ​ Connect Four One of the most famous strategy games in the world. Buy on Amazon ​ Rainbow Puzzle Ball A color matching game for kids. You push the colored balls around to match their color with the ring outside. Buy on Amazon ​ 3D Labyrinth Puzzles A toy for hands & eyes coordination and balance, spatial cognition, focus, observes ability and patience training. Buy on Amazon ​ Archemedes Puzzle Incredibly difficult and one of the oldest known puzzles and attributed to the great Archimedes. Buy on Amazon ​ Chinese Checkers Traditional Strategy Board Game with Set of 60 Colorful Marbles Buy on Amazon ​ Rock me Archimedes Suspense-filled balancing game that tests players’ strategic thinking. Buy on Amazon ​ Mondrian Blocks A game where math meets wıth Art Buy on Amazon ​ Mosaic Mysteries Puzzle Arrange mosaic tiles on a 2D plane in a way to make them seen as 3D. Buy on Amazon ​ Q-bitz Practice your symmetry, visual dexterity, quick thinking, spatial reasoning and memory skills. Buy on Amazon ​ The Genius Square STEM puzzle game with the combination of dice, location of the blockers Buy on Amazon ​ Tangram Puzzles Bubble Pop Tangram pieces with the Tangram Puzzles Book Buy on Amazon ​ Star Wars Chess Set A collectible - dark and light sides on the chess table. May the force with you. Buy on Amazon ​ Picasso Tiles Puzzle A 3D Puzzle promotes logic training, critical thinking, problem-solving, and hand-eye coordination skills. Buy on Amazon ​ Mancala Invented thousands of years ago, Mancala is one of the world's favorite games of counting and strategy. Buy on Amazon ​ Amazing Inventions Hands-on building projects that explore Da Vinci's invention ideas Buy on Amazon ​ Go - magnetic set Magnetic 19x19 Go Game Set Board Buy on Amazon ​ Crystallized A challenging board game for 2-4 players Buy on Amazon ​ Brain Games - I What is the most complex super computer;? The answer is really close to you! Buy on Amazon ​ ZomeTool - Kepler Cosmos German mathematician, astronomer Johannes Kepler's universe model made up of 5 platonic solids. Buy on Amazon ​ Snake Cube Fidget Snake puzzles can turn into any shape, you can combine them to create bigger shapes. Buy on Amazon ​ Number Slide The goal is to reposition the squares from a given arbitrary position by sliding them one at a time. Buy on Amazon ​ Hanoi Tower It repetitive sequential that allows moving one piece at a time and are only allowed to place a smaller piece on top of a larger piece. Buy on Amazon ​ ISS LEGO Ideas Series - International Space Station model Buy on Amazon ​ Battleship Introduction to Coordinate Plane and ordered pairs concepts Buy on Amazon ​ Color Code Challenge yourself wıth this visual perception game. Buy on Amazon ​ Genius Star Star version is even more challenging than the famous Genius square game Buy on Amazon ​ Tri-facta Multiplication Practice multiplication and division - a 3 people board game Buy on Amazon ​ Pattern Explorer 1 Book - diverse collections of pattern problems for students to explore, investigate, discover, and create. Buy on Amazon ​ String Art Create amazing geometric patterns - Set of 3 frames - 20, 30 and 40 points with frames of 15 cm diameter. Buy on Amazon ​ Quoridor Abstract strategy game for ages 8+ Buy on Amazon ​ Numbers & Letters Tangram Pattern Blocks Magnetic Jigsaw Puzzle with 24 Pcs Design Cards. Buy on Amazon ​ Magnetic Soma Cube A 3D Puzzle - Set of 7 Multi Shapes Magnetic Blocks with 54 Guide Cards. Buy on Amazon ​ Kanoodle - mini A 3D Puzzle Game, Over 200 Challenges Buy on Amazon ​ Geomag Magnetic Toys to build 3D Solids and explore 3D geometry Buy on Amazon ​ Buildables BUILD your own Spin Art Station with step-by-step instructions Buy on Amazon ​ Brain Games - II Book - full of puzzles, optical illusions, cranial challenges, and information on researches in neuroscience Buy on Amazon ​ The Shape-Shifting Box 3D Magnetic Transforming Magnetic Box Magic Cube - You can combine four of them to create other 3d shapes. Buy on Amazon ​ Hexagon n Puzzle Hexagon puzzle has many different solutions, Each card has hints for a different solution. Buy on Amazon ​ Soma Cube Each card has a building shape challenge on one side, and the solution on the other side. Buy on Amazon ​ Marble Run Marble Run for chain reactions by National Geographic Buy on Amazon ​ K'nex Levers & Pulleys Model- to build 3 unique lever or pulley models: a balance, a wheelbarrow and a Sailboat Buy on Amazon ​ We are a participant in the Amazon Services LLC Associates Program, an affiliate advertising program designed to provide a means for us to earn fees by linking to Amazon.com and affiliated sites.

  • Math Posters | MATH FAN

    Displays Math Boards Math Posters Math Class Floor Prints Math Cabinet Math Park MATH POSTERS Fun Math Fan Posters Free Download Spiral of Theodorus Free Download Sierpinski Triangle Free Download Ada Lovelace Free Download Pi Day Free Download Multiplication Table Free Download Iconic Number Free Download Marie Curie Free Download Square Root Day Free Download Star Wars - Astronomy I Free Download Kite Squares Free Download Measurements Free Download 17 Symmetry Groups Free Download Pyhthagorean Theorem Day Free Download World Oceans Day Free Download Star Wars - Astronomy II Free Download Binomial Cube Free Download Infinity Hotel Posters for Square Ceiling Tiles ​ Here is an example of the ceiling transformation. Click here for the prints of square ceiling tiles of the math classrooms. There are more than 60 different free posters in the file as a pdf document. ​ ​ Women in Mathematics (FREE) Poster Collection Free Posters Celebrating Women Role Models in Science, Technology, and Math by A MIGHTY GIRL . Visit her website to download these marvelous posters. Plus Posters Math Welcome Poster Banner Visit the great math blog M+A+T+H= love by Sarah Carter to see all the math posters she has created and download the Mathy Letters Poster Set created by + Plus Magazine (free) Mathematics Timeline Poster by Mathigon Visit Mathigon Gift Store to purchase the Timeline of Mathematics Poster Numberphile Poster Collection Visit Numberphile Merchandise to purchase one of a kind Math posters Fractal Posters Mandelmap Fractal Poster by Bill Tavis You can buy the poster in two different sizes from Amazon . To purchase: Here are some links where you can purchase mathematical posters; Nasco Education Tarquin Group Amazon

  • Projects | MATH FAN

    Math Fan Content Lessons Tasks Math Club Projects Math Project Ideas Math @ Home Games & Puzzles Math Magic Math & Art Long Term Projects, Researchs Mystery Mathematician "?" ​ Until 17th century algebra and geometry were two distinct branches of mathematics. He was the first man who combines algebra and geometry to provide a great tool for visualizing equations with two variables. Name this famous philosopher and mathematician. ​ His middle name is given to Coordinate Plane There are different stories about how he discovered the analytic geometry. Search about them. Introduce his findings and philosophy. Introduce the mathematical concept and equations. Write a few examples of his famous quotes. He is one of the other famous mathematicians who is also a very well-known Philosopher. Can you name others? What do you think, why are so many philosophers also mathematicians? Useful resources: 1 and 2 Back to Top Impossible Objects - How to make them possible? What is an impossible object? Do you see any impossible objects around you? -bottles with no insides (or outsides), one-edged loops, solid ball with no fixed size. Can you make any impossible objects by using only a piece of paper and glue? Have you ever heard about Mobius Strip? Who discovered it? Watch the video here and learn about the original name of the shape? Why this shape is so special? Repeat the same magical moves to create a four-twisted loop, a square and the hearts as you have seen in the video. Start your loop with a single twist. Make your prediction before unravelling the pieces–how many pieces will there be after halving horizontally? Will they be all the same size? How many twists will they have? Then cut it again. Answer the same questions. ​ What happens if you start with a double twist? ​ Then, search about the recycling logo. Who has designed it and when? Find out the origins of this logo. Useful resources: 1 Back to Top Mathematics and Philosophy What is mathematics? How do YOU define it? Is math science or art? Is math invented or discovered? Read the book “Is God a Mathematician by Mario Livio ” Watch the TEDed video by jeff Dekofsky. Find the famous mathematicians and their supporting arguments for these famous debates Write your thoughts and give examples to support your ideas. What do you think, why are so many philosophers also mathematicians? You can suggest additional sources to your readers to follow your footsteps. I mage is taken from the Authors' website https://www.mario-livio.com/books Back to Top Infinity and Far Beyond What is infinity? Give examples to infinite things? Can you make operations with infinity? How many natural numbers are there? How many evens? How many rational numbers are there between 0 and 1? What about between 0 and 2? So is one infinity bigger than another? Search about famous mathematician Cantor and his approach on the cardinality of the number sets What is the history of infinity? Invented or Discovered? Who has found its symbol? Search about Hilbert’s famous infinity hotel problem? Ask it to your friends. Suggested reading: Beyond Infinity by Eugenia Cheng Math and Infinity by Ali Nesin Back to Top Do prime numbers have primary importance? Is 1 a prime number? Are there more composite or prime numbers between 1-10? what about without boundaries? Is there a pattern among the prime numbers? Interesting facts about primes? ( ex: between a number and its double there is always a prime number) ​ List the methods to find primes. What is your favorite? Search about Goldbach Conjecture. Explain it by giving examples What are the other famous conjectures and theorems about prime numbers? ​ Visit https://www.mersenne.org to join internet’s biggest Mersenne Prime Search. ​ What is the largest known prime? Who , when and how was it found? ​ Watch the videos of Standupmaths videos by Matt Parker about prime numbers on YouTube. ​ Where do we use prime numbers in our daily life? Why are they so important? Back to Top Euclidian Geometry... Wait! There are other geometries?! Search about the origins of the Geometry? Who can be named as the father of geometry? Search about the plane Geometry? What are the basic axioms of plane geometry? Most of the Ancient Philosophers were also great mathematicians who studied the basic concepts of Geometry. Name a few of them. Additional search: focus on famous painting of Raphael “The School of Athens” give information about the mathematicians pictured in this masterpiece. Give examples of famous quotes about the relation between philosophy and geometry. Who has found the non-euclidian geometry? Find some demonstrations on internet Bring a spherical balloon or ball to your presentation to make a demo of non-Euclidian Geometry. Euclid's Elements Image is taken from the taschen.com Back to Top Iconic Number of Math It is not possible to write it as a ratio of two integers yet it is the ratio circle’s circumference to its diameter. People have calculated the first 10 trillion digits of it, though for most purposes – such as designing a building or sending a spacecraft to Mars. Search about the different calculation methods of this number ​ Calculate it by using infinite series Use Buffon’s Needle Problem Use the polygons method of the Ancient Greek mathematician Archimedes By playing Dart (1) Or with a pendulum (2) Look up the most famous rivers on earth. Calculate the ratio of the river's actual length to the distance from its source to its mouth as the crow flies. Ready to be surprised! You can find yourself discussing whether math is an invention or discovery! (3) Check the website & video for the artistic perspective of this fascinating number. There are several books, articles and online resources about this iconic number of Math. Complete your research and represent the most exciting facts about it. ​ Useful Resouces: 1 . 2 . 3 ​ Back to Top

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