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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 Books | MATH FAN

    Mathfan Shop Read. Watch. Play. Explore. Create Books Young Readers Mathflix P+ Games Toys and Gadgets 3D Models MATH & SCIENCE BOOKS Inspiration is calling! Math Games with Bad Drawings: 75 1/4 Simple, Challenging, Go-Anywhere Games―And Why They Matter by Ben Orlin This ultimate game chest draws on mathematical curios, childhood classics, and soon-to-be classics, each hand-chosen to be (1) fun, (2) thought-provoking, and (3) easy to play. Buy on Amazon The Complete Guide to Absolutely Everything (Abridged): Adventures in Math and Science by Adam Rutherford, and Hannah Fry Hannah Fry's Numberphile video about the book ​ Buy on Amazon Thinking Better - The Art of the Shortcut by Marcus Du Sautoy His interview about the book ​ Buy on Amazon Chasing Rabbits by Sunil Singh His Blog Post about the book Buy on Amazon Infinite Powers by Steven Strogatz Steven Strogatz’s brilliantly creative, down‑to‑earth history shows that calculus is not about complexity; it’s about simplicity. It harnesses an unreal number—infinity—to tackle real‑world problems, breaking them down into easier ones and then reassembling the answers into solutions that feel miraculous. Buy on Amazon The Great Unknown by Marcus Du Sautoy The Great Unknown challenges us to consider big questions—about the nature of consciousness, what came before the big bang, and what lies beyond our horizons—while taking us on a virtuoso tour of the great breakthroughs of the past and celebrating the men and women who dared to tackle the seemingly impossible and had the imagination to come up with new ways of seeing the world. Buy on Amazon A new kind of Science by Stephen Wolfram Wolfram uses his approach to tackle a remarkable array of fundamental problems in science, from the origins of apparent randomness in physical systems to the development of complexity in biology, the ultimate scope and limitations of mathematics, the possibility of a truly fundamental theory of physics, the interplay between free will and determinism, and the character of intelligence in the universe. Buy on Amazon A Mathematician's Apology by G. H Hardy G. H. Hardy was one of this century's finest mathematical thinkers, renowned among his contemporaries as a 'real mathematician ... the purest of the pure'. This 'Apology', written in 1940, offers a brilliant and engaging account of mathematics as very much more than a science; when it was first published, Graham Greene hailed it alongside Henry James's notebooks as 'the best account of what it was like to be a creative artist'. Buy on Amazon Geometry Snacks by Ed Southell and Vincent Pantaloni The idea behind the book is to show that problems can be solved in several ways, which means that, say the authors: “once a puzzle is solved, there are further surprises, insights and challenges to be had.. Buy on Amazon Cut the Knot by Alexander Bogomolny Cut the Knot is a book of probability riddles curated to challenge the mind and expand mathematical and logical thinking skills. First housed on cut-the-knot.org, these puzzles and their solutions represent the efforts of great minds around the world. Buy on Amazon Alan Turing: The Enigma by Andrew Hodges The Enigma, the story of the British computer pioneer and codebreaker Alan Turing. Critically acclaimed at the time — Donald Michie in New Scientist called it ""marvellous and faithful"" — the book was chosen by Michael Holroyd as part of a list of 50 'essential' books Buy on Amazon Thinking Classroom by Peter Liljedahl Sparked by observing teachers struggle to implement rich mathematics tasks to engage students in deep thinking, Peter Liljedahl has translated his 15 years of research into this practical guide on how to move toward a thinking classroom. Buy on Amazon Math Recess by Sunil Singh and Dr Christ Brownell In the theme of recess, where a treasure chest of balls, ropes, and toys would be kept for children to play with, this book holds a deep and imaginative collection of fun mathematical ideas, puzzles, and problems. Buy on Amazon God created the integers by Stephen Hawking This extensive anthology allows readers to peer into the mind of genius by providing them with excerpts from the original mathematical proofs and results. It also helps them understand the progression of mathematical thought and the very foundations of our present-day technologies. Buy on Amazon The Princeton Companion to Maths by Timothy Gowers This is a one-of-a-kind reference for anyone with a serious interest in mathematics. Edited by Timothy Gowers, a recipient of the Fields Medal, it presents nearly two hundred entries, written especially for this book by some of the world's leading mathematicians. Buy on Amazon Pi of Life by Sunil Singh Blending classic wisdom with over 100 pop culture references, Singh whimsically switches the lens in this book from the traditional society teaching math to a new and bold math teaching society. With charming buoyancy and intimacy, he takes us on an emotional and surprising journey through the deepest goldmine of mathematics-our personal happiness. Buy on Amazon Math with Bad Drawings by Ben Orlin Ben Orlin reveals to us what math actually is; its myriad uses, its strange symbols, and the wild leaps of logic and faith that define the usually impenetrable work of the mathematician. Buy on Amazon The Mathematics of Love by Hannah Fry Her TED talk on mathematics of love Buy on Amazon The Joy of x by Steven Strogatz The Joy of x, Steven Strogatz expands on his hit New York Times series to explain the big ideas of math gently and clearly, with wit, insight, and brilliant illustrations. Buy on Amazon Finding Moonshine by Marcus Du Sautoy This is the story of how humankind has come to its understanding of the bizarre world of symmetry – a subject of fundamental significance to the way we interpret the world around us. Buy on Amazon Uncle Petros and Goldbach Conjecture by Your Text Here​​ Uncle Petros is a family joke - an aging recluse in a suburb of Athens, playing chess and gardening. His young nephew soon discovers his uncle was once a celebrated mathematician who staked all on solving the problem of Goldbach's Conjecture. Buy on Amazon Things to make and do in the 4D by Matt Parker Things to Make and Do in the Fourth Dimension: A Mathematician's Journey Through Narcissistic Numbers, Optimal Dating Algorithms, at Least Two Kinds of Infinity, and More Buy on Amazon The Elements of Euclid by Oliver Byrne Nearly a century before Mondrian made geometrical red, yellow, and blue lines famous, 19th century mathematician Oliver Byrne employed the color scheme for the figures and diagrams in his most unusual 1847 edition of Euclid's Elements. Buy on Amazon Math Art by Stephen Ornes The worlds of visual art and mathematics come together in this spectacular volume by award-winning writer Stephen Ornes. He explores the growing sensation of math art, presenting more than 80 pieces, including a crocheted, colorful representation of non-Euclidian geometry that looks like sea coral and a 65-ton, 28-foot-tall bronze sculpture covered in a space-filling curve. Buy on Amazon Why study Mathemativs by Vicky Neale .. Buy on Amazon Women in Science by Rachel Ignotofsky A charmingly illustrated and educational book, New York Times best seller Women in Science highlights the contributions of fifty notable women to the fields of science, technology, engineering, and mathematics (STEM) from the ancient to the modern world Buy on Amazon The Math Book by Clifford A. Pickover Math's infinite mysteries unfold in this paperback edition of the bestselling TheMath Book. Beginning millions of years ago with ancient “ant odometers” and moving through time to our modern-day quest for new dimensions, prolific polymath Clifford Pickover covers 250 milestones in mathematical history. Buy on Amazon The Colossal Book of Short Puzzles by Martin Gardner The Colossal Book of Mathematics, have been selected by Gardner for their illuminating; and often bewildering; solutions. Filled with over 300 illustrations, this new volume even contains nine new mathematical gems that Gardner, now ninety, has been gathering for the last decade. Buy on Amazon Mathematics Magic and Mystery by Martin Gardner Why do card tricks work? How can magicians do astonishing feats of mathematics mentally? Why do stage "mind-reading" tricks work? As a rule, we simply accept these tricks and "magic" without recognizing that they are really demonstrations of strict laws based on probability, sets, number theory, topology, and other branches of mathematics. This is the first book-length study of this fascinating branch of recreational mathematics. Buy on Amazon Humble Pi by Matt Parker Matt Parker shows us the bizarre ways maths trip us up, and what this reveals about its essential place in our world. Mathematics doesn't have good 'people skills', but we would all be better off, he argues, if we saw it as a practical ally. This book shows how, by making maths our friend, we can learn from its pitfalls. Buy on Amazon Hello World by Hannah Fry Hannah Fry takes us on a tour of the good, the bad and the downright ugly of the algorithms that surround us. In Hello World she lifts the lid on their inner workings, demonstrates their power, exposes their limitations, and examines whether they really are an improvement on the humans they are replacing Buy on Amazon Is God a Mathematician? by Mario Livio Explores the plausibility of mathematical answers to puzzles in the physical world, in an accessible exploration of the lives and thoughts of such figures as Archimedes, Galileo, and Newton. Buy on Amazon Logicomix by Your Text Here​​ This exceptional graphic novel recounts the spiritual odyssey of philosopher Bertrand Russell. In his agonized search for absolute truth, Russell crosses paths with legendary thinkers like Gottlob Frege, David Hilbert, and Kurt Gödel, and finds a passionate student in the great Ludwig Wittgenstein. Buy on Amazon Limitless Mind by Jo Boaler In Limitless Mind, she explodes these myths and reveals the six keys to unlocking our boundless learning potential. Her research proves that those who achieve at the highest levels do not do so because of a genetic inclination toward any one skill but because of the keys that she reveals in the book. Buy on Amazon Fermat's Last Theorem by Simon Singh In 'Fermat's Last Theorem' Simon Singh has crafted a remarkable tale of intellectual endeavour spanning three centuries, and a moving testament to the obsession, sacrifice and extraordinary determination of Andrew Wiles: one man against all the odds. Buy on Amazon How to bake Pi by Eugenia Cheng In How to Bake Pi, math professor Eugenia Cheng provides an accessible introduction to the logic and beauty of mathematics, powered, unexpectedly, by insights from the kitchen: we learn, for example, how the béchamel in a lasagna can be a lot like the number 5, and why making a good custard proves that math is easy but life is hard. Buy on Amazon The Wonder Book of Geometry by David Acheson David Acheson takes the reader on a highly illustrated tour through the history of geometry, from ancient Greece to the present day. He emphasizes throughout elegant deduction and practical applications, and argues that geometry can offer the quickest route to the whole spirit of mathematics at its best. Buy on Amazon The Nature of Mathematics by Karl J. Smith Karl Smith introduces you to proven problem-solving techniques and shows you how to use these techniques to solve unfamiliar problems. best. Buy on Amazon A History of Mathematics by Victor J. Katz A History of Mathematics, 3rd Edition, provides students with a solid background in the history of mathematics and focuses on the most important topics for today’s elementary, high school, and college curricula. Buy on Amazon The Simpsons and their Mathematical Secrets by Simon Singh Simon Singh reveals, underscores the brilliance of the shows' writers, many of whom have advanced degrees in mathematics in addition to their unparalleled sense of humor. Buy on Amazon Beyond Infinity by Eugenia Cheng Beyond Infinity: An expedition to the outer limits of the mathematical universe Buy on Amazon Tales of Impossibility by David Richeson Tales of Impossibility recounts the intriguing story of the so-called problems of antiquity, four of the most famous and studied questions in the history of mathematics. Buy on Amazon Math without Numbers by Milo Beckman Math Without Numbers is a vivid, conversational, and wholly original guide to the three main branches of abstract math—topology, analysis, and algebra—which turn out to be surprisingly easy to grasp. Buy on Amazon Change is the only Constant by Ben Orlin Change is the Only Constant is an engaging and eloquent exploration of the intersection between calculus and daily life, complete with Orlin's sly humor and memorably bad drawings. Buy on Amazon 1089 + all that by David Acheson David Acheson's extraordinary little book makes mathematics accessible to everyone. From very simple beginnings he takes us on a thrilling journey to some deep mathematical ideas. On the way, via Kepler and Newton, he explains what calculus really means, gives a brief history of pi, and even takes us to chaos theory and imaginary numbers. Buy on Amazon Do Dice Play God? by Ian Stewart From forecasting, to medical research, to figuring out how to win Let's Make a Deal, Do Dice Play God? is a surprising and satisfying tour of what we can know, and what we never will. Buy on Amazon Mathematics for Human Flourishing by Francis Su or mathematician Francis Su, a society without mathematical affection is like a city without concerts, parks, or museums. To miss out on mathematics is to live without experiencing some of humanity’s most beautiful ideas. 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.

  • MATHFLIX P+ | MATH FAN

    Mathfan Shop Read. Watch. Play. Explore. Create Books Young Readers Mathflix P+ Games Toys and Gadgets 3D Models MATHFLIX PRIME +

  • 3D Printed Math | MATH FAN

    Mathfan Shop Read. Watch. Play. Explore. Create Books Young Readers Mathflix P+ Games Toys and Gadgets 3D Models 3D PRINTED MATH If you have access to a 3D Printer, try printing toys, gadgets, puzzles, and math manipulatives. ​ Creativity is a major part of growing up, and 3D printed toys can let kids express that. Everyone can design 3D models to print or visit websites like Thingiverse to find free models to customize or print directly. You can support the designer by clicking the tip the designer button. Do not worry that if you cannot 3D print, you can still purchase the one-of-a-kind 3D models. Out of gallery Fractals by henryseg Hilbert Curve One of the models mentioned in the Visualizing Mathematics with 3D Printing Book. You can use it to display iterations of HC View More Topology by henryseg Topology Joke One of the models mentioned in the Visualizing Mathematics with 3D Printing Book by Henry Segerman View More Pyramids & Cone by mrbenbritton Height vs Slant Height A set of pyramids and a cone which have had slices taken out to help students identify the hidden triangles. View More Knots by mathgrrl Knots 3D printed conformations of the fifteen knots through seven crossings View More Scutoids by mathgrrl Pair of Packable Scutoids Scutoids are a geometrical solution to 3D packing of epithelia in the journal Nature. View More Cone by IronOxide Conic Sections A horizontal slice reveals the circle, a slanted reveals the ellipse, a vertical reveals the hyperbola, and a slanted vertical cut reveals the parabola. View More Pythagorean Thm by Freakazoid Proof of the Theorem Use the given regions to cover the squares to prove Pythagoras Theorem ​ ​ View More Illusion by KazukiYamamichi Straight vs Curve Straight bar goes through curved slit ​ View More Klein Bottle by MadOverlord Klein Bottle This special 3D shape with only one surface. ​ ​ View More Reuleaux Shapes by roklobster04 Solids of Constant Width These shapes have the same width (height) no matter how they are rotated or rolled. View More Puzzle by Juanill0 Puzzle 365 Famous Puzzle 365 - use petrominoes to cover all the numbers except today's date. View More Fidgets by JustinSDK Curves of Pursuits 3D Models of fidgets which can also be used to display the curves of pursuit. ​ View More Sierpinki Pyramid by ricktu 3D Fractal Watch the horizontal cross-sections when forming the pyramid. ​ View More Pentagons by mathgrrl All Tessellating Pentagons 15 types of non- regular tessellating pentagons View More Platonic Solids by XYZAidan Foldable Polyhedra oldable versions of your favorite regular 3D shapes, the five Platonic Solids! View More Menger Slices by mathgrrl Slices of Menger Cube Three different diagonal slice models of a level 2 Menger sponge. ​ ​ View More Two Circle Roller by AJsRaceway Oloid-like shape When it rolls, every point on its surface touches the ground ​ View More Hinged Square by jonco223 Square & Triangle Transformation inged triangle-square dissection (Dudeney's dissection) ​ View More Mobius Strip by dennedesigns Mobius Strip 5 half-twists 2 sided but one surface special shape View More Impossible Triangle by Tomonori 3D Optical Illusion A triangle with three 90 degrees angle View More Inclinometer by ChrisX35 Hypsometer Usually called inclinometer. It is used to find the height of the objects using trigonometry. View More Pyramids & Cube by ereiser Volume Formula Can be used to prove the volume ratio between a cube and a square pyramid. ​ View More Cipher by ereiser Cryptography Students learn about the Caesar Cipher. They practice encoding and decoding messages using a decoder ring. View More Brachistochrone by Taevinrude Brachistochrone Curve Demonstration Which is the fastest possible path a ball can take when falling between two points? View More Trammel by lgbu Trammel of Archimedes A geometric structure for constructing ellipses or exploring mathematical relations View More (A+B)^3 by eashwarps Binomial Expansion A model for the binomial expansion of the cube of the sum of a and b. ​ ​ View More Illusion by dietervdf Perspective 3D object w/ the top view is a circle, the front view is a triangle and the side view is a square. View More 64=65 by althepal Slope Puzzle 64=65 Rearrange the pieces, and the area seems to change. How can that happen? View More Squarcle by PhilKloppers 3D Optical Illusion he Squarcle is a 3D optical illusion based on the work of Kokichi Sugihara View More Golden Ratio by stephenplace Golden Ratio Spiral A key ring of Fibonacci - Golden Ratio spiral View More Book Recommendation; Visualizing Mathematics with 3D Printing by Henry Segerman

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