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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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  • 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 Collect 100 Free Download Collatz Conjecture Free Download Goldbach Conjecture Free Download Spiral of Theodorus Free Download Pyhthagorean Theorem Day Free Download World Oceans Day Free Download Star Wars - Astronomy II Free Download Binomial Cube Free Download Infinity Hotel Free Download Egyptian Fractions Free Download Number of Triangles Free Download String Art Free Download Katherine Johnson Free Download Sierpinski Triangle Free Download Ada Lovelace Free Download Pi Day Free Download Multiplication Table Free Download Iconic Number Free Download Notice & Wonder Free Download Similar & Different Free Download Fun Facts - Pi 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 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

  • Math Cabinet | MATH FAN

    Displays Math Boards Math Posters Math Class Floor Prints Math Cabinet Math Park MATH CABINET Cabinet of Wonders ​ We all know mathematics is a very colorful subject. It had many wonders so why not share some of the examples in a math cabinet? The main aim here is to give students a fun, interesting, social, and tactile experience of fundamental math concepts. Sometimes students can also design the displays and materials; in this way, they will develop ownership over the subject and the classroom they live in every day. ​ Maths and Art Projects and Activities Click here to go to the MATH & ART page to see a large selection of maths & art activities, ideas, lesson plans, templates.. Math Class Essentials Pattern Blocks Wooden Building Blocks Geometry Sets Prime Climb Game Geometiles 3D Building Set Grid Boards Unit Cubes Nets of 3D Solids Origami Geoboords Fractions For more... Books A list of Math and Science Books for readers of all ages. Ready to be amazed and love math even more and get inspired! Books for Young Readers Math and Science are full of great stories, myths, and legends as well as new adventures. Let young readers learn about the other side of the coin! Toys and Gadgets People are more likely to learn math principles when surrounded by relevant concepts. And the best part is they are GENDER-FREE!​ Puzzles and Games Who would not like playing games? Games are yoga for your mind. Play games to explore more, solve problems, improve systematic and strategic thinking. 3D Print Templates Printing can make the toys cheaper and extremely customizable. Creativity is a major part of growing up, and 3d printed toys can let kids express that.

  • Puzzles and Games | MATH FAN

    Mathfan Shop Read. Watch. Play. Explore. Create Books Young Readers Mathflix P+ Games Toys and Gadgets 3D Models Puzzles and Games 21st Century Pattern Blocks Explore Eight different shapes and colors for endless possible creations by Math for Love Buy on Amazon ​ Talking Math with Kids Shop Buy tiling turtles, Truchet Blocks, and Christopher’s books Which One Doesn’t Belong? and How Many? Buy on Talking Math with Kids Shop ​ Turing Machine Preorder today! Expected release November 16, 2022! Buy on Labyrinthdc ​ 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 ​ The Genius Square Game of The Year Award Winner! Roll The Dice & Race Your Opponent to Fill The Grid! Buy on Amazon ​ Rock me Archimedes Suspense-filled balancing game that tests players’ strategic thinking. 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 ​ Prime Climb Explore mathematical structure in multiplication, division, and prime numbers by Math for Love Buy on Amazon ​ Chain Reaction Sets by Zig Go Build different layouts and tracks, then start the action and watch the reactions. 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 ​ 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.

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