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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: ------ ------ ------

## Other Pages (42)

- Math Competitions | MATH FAN
COMPETITIONS One of the best parts of the math-fests is obviously being able to run lots of different competitions for the kids who have different ability levels, skills, and strengths. Our aim to have so many competitions is to reach everyone at the school. You do not need to be among the top kids of the school to participate in one of those competitions. Just join, learn, collaborate, and have fun! For further information on competitions, the KocMathTeam website will be online soon ...

- Math at Home | MATH FAN
Fun Mathfan Content Lessons Tasks Math Club Projects Math @ Home Math @ Home Games & Puzzles Math Magic Math & Art Everyday Math Day 1: Roman Arch Bridges Grab all the cushions, books or Jenga blocks at home and try to build an arch bridge. The forces of a Roman arch so strong that arches can stand without any glue or other adhesive holding them together. Try it for yourself! How it works: Its semicircular structure elegantly distributes compression through its entire form and diverts weight onto its two legs, the components of the bridge that directly take on pressure. Roman Bridge, Ponte da Vila Formosa, Portugal Blueprint of the Arch Bridge Home Made Arch Bridge Roman Bridge, Ponte da Vila Formosa, Portugal 1/3 Image attributions: https://www.ancient.eu/image/4407/roman-bridge-ponte-da-vila-formosa-portugal/ https://www.thisiscarpentry.com/2012/01/06/circular-based-arches-part-1/ https://www.thelistlab.net/blog/how-to-make-a-book-arch Resources: https://kids.nationalgeographic.com/explore/books/make-this/roman-ice-arch/ Day 2: Leonardo Da Vinci’s Famous Self-Supporting Bridge Do you have popsicles at home? I did not try with toothpicks or q-tips, but I think that they may also work. Other than those, “Patience” will be the main thing you will need. Leonardo Da Vinci’s Self-Supporting Bridge is also known as the emergency bridge. No nails, screws, rope, glues, notches, or other fasteners are holding the bridge in place. You can also watch the step by step instruction video but first I suggest you try by looking at the image below. How it works: You will be weaving the sticks together so that the tension between the sticks keeps the bridge together and lifts it off of the ground. You may also watch the video on YouTube how a father and son build the bridge at their backyards to motivate yourself to keep going :) 1/4 Image Attributions: https://www.core77.com/posts/65043/Leonardo-da-Vincis-Ingenious-Design-for-a-Self-Supporting-Bridge Resources: https://thekidshouldseethis.com/post/how-to-make-leonardo-da-vincis-self-supporting-arch-bridge https://www.instructables.com/id/Da-Vinci-Popsicle-Stick-Bridge/ Day 3: Cylindrical Mirror and Anamorphic Art The original is the usage of mirroring paper, but nowadays unwrinkled aluminum foil can be used as well (But because the images are fuzzier, the observations may not be as clear.) And a soda or coke can, or any cylindrical object that you can cover with the aluminum foil is ok. After you create the cylindrical mirror, you may either color an already distorted image (1) or print the polar grid (2) below and create your own anamorphic art. How it works: Making anamorphic drawings involves mechanically distorting an image by transferring the image from the square grid (the original image) onto a polar grid (distorted grid. It is a mapping, or a correspondence, between a cartesian set of coordinates, and a polar set of coordinates. Place your cylindrical mirror on the circle and look into the mirror to see the image restored. Resources: https://anamorphicart.wordpress.com/2010/04/21/cylindrical-mirror-anamorphoses/ https://raft.net/wp-content/uploads/2019/03/278-Anamorphic-Art.pdf https://makezine.com/projects/draw-distorted-pictures/ Coloring a Distorted Image (1) You may color the distorted image below from makezine.com, in that case, my sure that the cylindrical shape you will find at home matches the circle at. The center of the paper. Link for already distorted Makey Bot image : Polar Grid Template (2) You can also use this polar grid by printing to make your own drawings; Link for the Polar Grid Day 4: Pi at home You can do lots of different pi activities at home. I want to list a few very popular ones. Pi – skyline: All you need is paper, ruler and crayons, create black bars at the lengths of the digits of Pi and create the skyline for Pi-York, Pi-ris, Pi-lan, Pi-chester …. Building Pi-City with Lego: Instead of coloring the digits of Pi, you can use the lego pieces to actually build your Pi-city Pi – bracelet: If you have the toolkit, all you need is to give each color a number like; 1 -pink, 2-blue, 3-green, 4-red .. And you can start forming your pi bracelet. Pi Art in a Circle: Simply divide a circle into 10 equal intervals label them from 0 to 9 if possible each with different colors. Start drawing lines from 3 to 1, 1 to 4, 4 to 1 and go on … Use the same color for the segment with your starting point for each of the drawings … Pi – Dart Game If You have Dart Board at home by throwing dart, you can calculate Pi. Here all you can do is watching this video. Before I forget, Pi number; 3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 ... Image Attributions and for more information please visit; https://www.whatdowedoallday.com/ http://www.pinkstripeysocks.com/2014/03/pi-day-activity-make-pi-day-bracelets.html Day 5: String Art Another paper, pencil and ruler only activity. But this is to create your art on the paper. If you want to create some 3d art, you can always use a corkboard, pins, and some string. Even if you have the necessary materials for 3d art, I recommend you start with paper and pencil first. Draw a big “L” shape on a paper and mark the numbers with equal intervals till 15. Or you can use the templates below. Again, all you are gonna do is drawing straight lines with a ruler to connect the points such as the; The first point on y-axis goes to the last point on x-axis The second point on y-axis goes to the second-last point on x-axis … Spoiler Alert: When you have finished you’ll see that you have created a curve by using straight lines. You can extend your initial drawing by converting your L shape to a “+” plus sign Then you can try a 60 angle “<” as your initial figure and complete it to a hexagon by connecting 6 of them from their corners. (Here you can use less number of points on the lines..) String art is a topic with no limits if you feel like you are interested, make sure you’ll make an internet search. Have math fun... 1/9 Links for the String Art Templates: L Shape + Shape Square Shape Octagon 60 degrees Hexagon 1 Hexagon 2 Day 6: Vedic Worms Fill in the multiplication table grid and reduce double-digit numbers to a single digit by adding the digit of the products. Example: If 9×9=81, add the numbers in the sum (8+1), and put the sum of 9 in the square. If the new sum is also double-digit, add those numbers. Example: 7×8=56; 5+6=11; 1+1=2. Place the number 2 in that square. We are going to use this number sequences to create the Vedic Worms which are also spirolaterals Spirolaterals are geometrical figures formed by the repetition of a simple rule. The pattern is formed by drawing line segments of a certain length from a number sequence with a fixed angle and a direction. Although the spirolaterals can be created with any number sequence, we will use the Vedic Squares we have created. That’s why they are also called “VEDIC WORMS”. Start with a row of numbers you choose. (1,2,3,4,5,6,7,8,9) These numbers will determine the length of each ‘step’ of the ‘spirolateral’. 2. Choose a direction; clockwise (CW) or counterclockwise (CCW) 3. Choose a grid type to draw on (In fact here, you are choosing the angle of your movement) Square Grid (90 degrees ) Isometric Grid (60 degrees ) Hexagonal Grid (120 degrees ) … 4. Now start drawing spirals through your list. For example, if we choose CW direction on a square grid with the first row of numbers, It means 1 step up, 2 steps right, 3 steps down and 4 steps left then repeat like 5 steps up, 6 steps right, 7 steps down and 8 steps left and 9 steps up to complete your drawing. Please check t he post about the Vedic Squares, Worms and The Spirolaterals for the necessary materials.. Day 7: Two (Hinged) Mirrors and Shapes If you have two small mirrors at home, it means you are ready for this activity. I had two hinged mirrors that I got from Amazon recently, but any mirrors like some of the foldable vanity mirrors or small Ikea ones will do. In addition to the pair of mirrors, any shapes, tangram pieces, lego pieces, different shaped toys can be used for this activity. 1. Angles, Reflection, Tessellation: Tessellation is covering a surface with a shape(s) without any gaps or overlaps. You can create your tessellation by using lego pieces, shapes anything you can create your design. Then, use the mirrors to enlarge your design: Arrange the mirrors as a straight line (180) to double your design! Arrange the mirrors with a 120-degree angle in between to triple your design! Arrange the mirrors with a 90-degree angle in between to ? your design! Arrange the mirrors with a 60-degree angle in between to ? your design! 2. Lego Pieces, Other Half, Polygons Create a car, a space ship, a dragon whatever you like but only half of it, then use the mirror the create the other half. You can do the same by holding the mirrors with different angles to enlarge your designs. Now use a thin, long lego piece, or any toy that you can use as a line segment. Arrange your mirrors with a 120 degrees, put the lego piece in between, what is the name of the polygon you have created? Try the other angles (You can measure the angles with a protractor) what kind of polygons you can create? What if you want to form a polygon with 12 sides (dodecagon), how are you going to arrange the mirrors? You can repeat the same activity by drawing a line segment on a paper and putting the mirrors on it by creating different angles between them. 3. Fractions and Creativity Let's try something else, if you have two identical triangles like tangram pieces ( if don't simply draw, color and cut two identical triangles from paper) arrange them in all the possible ways to create a square by using the mirrors? Which angle you need to use to create the square? How many different designs can you make? What fraction of your design is purple? What about your initial shape? Are those fractions equal? How many different ways you can divide a square into halves? If you want to create a snowflake with a shortcut, what would be the angles between your mirrors? If you want to draw an octopus by drawing only one of its arms, then which angle you need to use? BY USING TWO MIRRORS, YOU CAN LEARN ABOUT; ANGLES POLYGONS TESSELLATION SYMMETRY REFLECTION FRACTIONS ..

- Museums | MATH FAN
S.T.E.A.M Museums EXPERIENCE SCIENCE IN ACTION... It is nearly impossible to exit a museum without having gained any information or insight. A single visit to a museum can expose the visitors to in-depth information on a subject and a great inspiration for their studies in the future. These informal learning experiences deserve more attention from educators and parents. Without the stress of assessment and restrictions of strict curriculums, they can transform people's math&science-related perceptions by providing inquiry-based and hands-on experiences, explorations, and individual discovery. I believe giving kids space to explore what they are interested in is the most effective method of motivation for math & science. AND If you are a parent planning to visit the cities below, make sure your children also have time and opportunity to see these mind-blowing museums and science centers. You would not believe the change in their perspective towards science and math while having so much fun. Istanbul, Turkey Rahmi M. KOÇ Museum San Francisco, US California Academy of Sciences San Francisco, US Exploratorium Los Angeles, US Griffith Observatory New York, US Momath London, UK Science Museum Rome, Italy Explora Florence, Italy Galileo Museum Wolfsburg, Germany Phaeno Museum Frankfurt, Germany Mathematikum Munich, Germany Deutsches Museum Greece Plato's Academy Acropolis and Parthenon Museum Herakleidon Pythagorio, Island of Samos Figueres, Spain Dali Theatre - Museum Mosaics Museums, Turkey Antep Zeugma and Hatay Archaeology Museums Den Haag / Holland Escher Museum Anchor 1 Anchor 2 Anchor 3 It is a regardless fact that limited resources are a big obstacle for off-campus visits. Here, Google Arts & Culture gives us a miracle solution. One can find lots of inspiring museums, galleries all over the world. Removing the boundaries of the accessibility of such an important resource through the help of technology is a very valuable skill for all teachers to have. Here is a l ink to the post to learn more about Google Arts & Culture Click here for a short presentation about Math Museums. ppt , pdf