Truncated Hexagonal Pyramid in Cabri 3D

You can find the steps of truncated hexagonal pyramid construction in the following link. https://www.youtube.com/watch?v=qyz0hOC49H0&t


This is the final state of my construction, you can download it from the following link and open it in your personal computer with Cabri 3D. https://drive.google.com/file/d/1rUxZpPxpCGbVCidAKRJB_Z0Eq1PKb5lU/view

Math Playground

     There is an issue that I want to write for a long time. Educational games. I want to touch upon the importance of games in children’ lives, we all realize this effect. I think that one of the easy ways of loving mathematics for children is computer games which kids cannot say no. What we need to do is determine how can we make this situation more useful for children. Integrating mathematics into games may be the best way to make their interests of games into something much more beneficial. By this way, educational process can be fun and appealing for children by overcoming the boringness of traditional classes.

     Let me introduce Math Playground for you. It is a website which you can reach the online mathematics games with good quality. In the website, there are some main parts such Math Games, Logic Games, Math Arcade, Story Math, and a lot of different categories. According to me, some of very good categories are Hidden Math Concepts, Visual Math Tools, and Brilliant Brain Workouts.

     For example, when you enter the category of Hidden Math Concepts, you can see the games separated by topics. 

     Let’s choose Reflect and Rotate game. You will see the screen like that. Game offers different opinions for you.

     Under the game screen, there is a table that indicate learning connections of game.

     The other method for categorizing the games in site is grade level. The content of the site covers topics from the first grade to sixth grade. When you enter the site, you can see the grade levels on the top of the page.

For more information, click here.

     These kinds of games allow the children to focus on the area of personal interest and improvement your problem solving and critical thinking skill when they have fun.

After a short time, I will share the writing which I mention my thoughts about game, educational game, and gamification.

     Take care of yourselves and do not forget the great influence of games on children 😊

PROJECT OF SQUARE ROOT SPIRAL

There is not one way to learn and teach in mathematics. We can teach the concepts by non directly way and through technology we have different effective software we can use for more comprehensive learning. For this reason, I want to share our project -it is square root spiral- with you. As you see,there are some objectives related to Turkish curriculum. Then you can see the pedagogical knowledge and how you can construct and use the square root spiral in our lessons.

Objectives

M.7.2.1.3. Sayı örüntülerinin kuralını harfle ifade eder, kuralı harfle ifade edilen örüntünün istenilen terimini bulur.

M.8.3.1.5. Pisagor bağıntısını oluşturur, ilgili problemleri çözer.

M.8.3.2.1. Nokta, doğru parçası ve diğer şekillerin öteleme sonucundaki görüntülerini çizer.
b) Dinamik geometri yazılımları ile yapılacak çalışmalara da yer verilebilir .

M.8.3.2.3. Çokgenlerin öteleme ve yansımalar sonucunda ortaya çıkan görüntüsünü oluşturur.

9.5.4.1. Dik üçgende Pisagor teoremini elde ederek problemler çözer.
a) Teorem elde edilirken model çeşitliliğine yer verilir.
c) Pythagoras’ın çalışmalarına yer verilir.

9.5.4.3. Dik üçgende dar açıların trigonometrik oranlarını hesaplar.
a) Bir açının sinüs, kosinüs, tanjant ve kotanjant değerleri dik üçgen üzerinde tanımlanır.
ç) Bilgi ve iletişim teknolojilerinden yararlanılır.

Pedagogical Knowledge

   This tool is constructed to introduce square root spiral. Square root spiral (also called Theodorus spiral, Einstain spiral or Pythagorean spiral) is a spiral composed of contiguous right triangles.

   When students think about the square root spiral, they will discover and explore that there is a relationship between the hypotenuses of contiguous right triangles in the spiral. In comparison the direct instruction, discovering by their inference gives opportunity to them to increase their higher-level thinking.

   To construct the square root spiral, they use more than one mathematical concepts together and connect with each other. First one is Pythagorean Theorem; students find the length of the hypotenuse by taking advantage of the lengths of upright edges. In square root spiral, the hypotenuse of the triangle will be one of the upright edges of the next triangle. The other concept is trigonometry; students use the trigonometrical ratio to find the angles of contiguous right triangles. Students also observe that if the measure of angle is getting smaller, the length of edge opposite the angle is also getting smaller.

   Students find the rule of pattern of spiral. To find the pattern, it is necessary to discover that decrease in angle is depend on the increase of the adjacent edge.

   In addition, students encounter the example of construction which is created by using iteration based on angles and lengths. Students in 8th grade have a chapter about transition, rotation, and reflection. Some students do not have the critical thinking ability, the imagination is hard for them. Thanks to square root spiral tool, they can easily see the transformations.

   So, the square root spiral is widely effective construction for students to practice more than one mathematical concepts together and to develop the critical thinking skills.

User Manual

   Teachers can use the Square Root Spiral as an activity.

   Firstly, teacher hand out two worksheets. In the first one, there is a square root spiral in the following picture, and in the second worksheet there is a table consist of some calculations which students should make by using the information in the picture.

   After students make these calculations, they will see the connection between the hypotenuse of contiguous right triangles and figure out the rule of pattern in square root spiral by using the information in the table.

   Finally, teacher show our tool to students to strengthen the learning because the visualization makes the information make permanent. They realize that if we change the length of first segment, then the lengths of all other segments change with the same number. So, we can generalize the square root spiral to any length of the segment.

• You can find steps of the square root spiral construction with different two ways in Geometer’s Sketchpad in the following link: https://www.youtube.com/watch?v=jif0CR0xJbU

• This is the final state of our construction, you can download it from the following link and open it in your personal computer with Geometer’s Sketchpad. https://drive.google.com/open?id=1VyObNFWvUIwP1yeUuclvZsMWl4bb5xs4 

Another Sofware: GEOGEBRA

   In my previous post, I have mentioned about Geometer’s Sketchpad. Now, I want to introduce GeoGebra for you. GeoGebra is a free dynamic software which can be used to learn and teach mathematics and geometry. Its name arises from geometry and algebra. Although there are a lot of similarities between Geometer’s Sketchpad and GeoGebra, when we compare them, GeoGebra has more complex structure. In addition, you can open a GeoGebra account yourself, share their constructions in there, and follow other users who you like their sharing.

   GeoGebra has different types which are GeoGebra Graph-Calculator, GeoGebra Geometry, GeoGebra 3D Graph, and Mathematics Applications. You can use one of them according to your aim.

   GeoGebra has been rapidly having popularity among teachers and researchers around the world, because it is easy-to-use dynamic mathematics software that combines many aspects of different mathematical packages. GeoGebra encourages students to make projects in mathematics and geometry since students can personalize their work in this software. For example, they can change the colors, font size, line style according to their preferences, they can make some constructions by different ways; there is no limitation in how they construct.

   You can design a game relevant to your topic by using GeoGebra. So, the topic become more interesting for students because today’s children spend most of their times with games and learn too much. The visualization is possible with GeoGebra for the student to see and explore mathematical relations and concepts which were difficult to show in past lessons without technology.

   I want to talk about reasons of why you can use this software. First, GeoGebra is a free open source dynamic mathematics software. So, you can freely download it from http://www.goegebra.org/. Because of that, you do not face the license problems. Secondly, GeoGebra is constantly being developed by various researchers. Its database is always updating and improving.

Geometer's Sketchpad and Euclidea

     Hi everyone,

     I want you to introduce ‘Geometer’s Sketchpad’. It is a dynamic software which increases the students’ interest of geometry and mathematics. We can provide the students to understand and comprehend geometry and mathematics more easily by the visual support thanks to GSP. You can do a lot of things which helps to use in class activities with class. You can create points, straight lines, circles, etc. on the screen. And then you can hide unnecessary point and do animation show.

     I have learnt the GSP in this semester. An instructor showed us some basic constructions and we tried to construct on our computers individually. If we did not find the way, we asked our friends for help and expressed our opinions collaboratively. In this way, we discovered a lot of properties of GSP. For example, when we tried to do equilateral triangle in sketchpad, we showed that there is more than one way for this. So, when the students study with this software, they can find different ways to do same things. In this way, make an interference and understand more deeply.

     You can find more information about GSP in http://www.dynamicgeometry.com/ .

     If you want to introduce GSP, this website might be useful. You will find several short tutorial videos which shows basic properties of GSP. https://www.atomiclearning.com/geomsketch_pc

     Another thing I have to mention about is a game whose name is Euclidea. In this game, there are some tasks which you have to complete to pass next level. Every level has average 9-10 tasks and you have to earn three stars from them. Solving the task is enough to earn one star. The main goal is to solve a problem with minimum number of moves. There are two moves: E is elementary Euclidean constructions and L is straight or curved lines. Each level has E and L goals. You have to complete these goals to earn another two stars. Earning three stars from all tasks is not enough to pass the next level, you have to find some hidden stars. To find them, you should indicate the all solutions at the same drawing. In generally, hidden stars occur in the symmetric situations.

     You can play this game in website https://www.euclidea.xyz/en/game/packs or download from App Store and Google Play.

     I hope, I can arouse you interest about this game 😊

3D Printer Technology for Education

        I want to mention about 3D printers which is not common as it should be in Turkey. There are many people who do not know yet. Especially in mathematics issues that students have difficulties to imagine, some basic shapes can be printed by using these printers. I want to share a good example of pattern which is made with 3D printer.

   Most of students generally have difficulties to imagine geometric shapes, this causes that students cannot understand the logic of issues deeply. They solve problems without understanding why they do, just from formulas and rules. Sometimes angles are considered as there are just on paper but there are in everywhere of real life. Students can realize this fact thanks to this printed model and visual learning provides permanent information because students realize the relationship between angles and shapes in real world themselves. In addition, this model helps the students to understand the concepts of an angle and measure of an angle. Students have also difficulties to discriminate between them, most of them think an angle as measure of angle. By this model, they can also see relationship between types of angles obviously because angles can be removed from their places. For example, they can see that addition of right angle and acute angle is smaller than straight angle by trying to motion the angles. So, they can compare the angle of addition and straight angle.

         In this website, you can find more example of projects which is done by using 3D printer. https://www.thingiverse.com/education Moreover, you can order these kinds of models from 3D printer centers. If you want to more information about 3D design products, one of my friends is interested in these kinds of things. They prepare the products in different size and colors depending your wishes. I can help you to contact him 😊

Technology Integration in Math

     Technology makes our lives easier in countless area such as education, transportation, telecommunication. I can say that everything which helps us to live and understand something is a part of technology. Technology is a something is always improving by the changing of living conditions. When I was a student in primary school, my teacher sometimes uses the cyclopes in lessons but now teachers use projection or something like these. We use the moodle system to share announcements; we can load our assignments to there so we do not have to take a printout thanks to moodle. Instructors use power point slides which is more effective than books for instruction in lectures. I think these kinds of things are all technology.

     In math education, using technology have also significant role. For example, I learn the derivative in high school, I solved so many questions in lessons and exams in these years. I thought that I am good at this subject. Some of my classmates cannot solve the problems about it, I also helped them. When I saw the definition of derivative at university, I cannot make sense of it. Then last week the teacher showed us the graphic about derivative which is made with Geometer’s Sketchpad. I really understood the what derivative is and where its definition comes from. If all teacher uses these kinds of programs in math lessons, number of the students who does not like mathematic might decrease. I think the reason of disliking mathematic is that students cannot understand and visualize some abstract things. By these kind of applications, mathematic could become more understandable and enjoyable for students who do not like it.

     Technology integration makes learning more permanent especially for primary school students. When they encounter three dimensional shapes, they might have difficulties about understanding and visualizing. Integrating technology in math is useful for teachers as much as student. One of the important advantages of technology integrated in instruction for teachers is that they can save the time to draw complicated shapes, graphs, etc. There is a vital point that should be mentioned. Using technology provide to save the time but it is used just when it is necessary. If we try to integrate it in every subject, it begins to be disadvantage of us in terms of time saving because every subject is not appropriate to use technology. It is the important to make this distinction. Another crucial point is that firstly teachers should learn how they use the technology in effective way. There are so many programs they can use but teachers cannot utilize from these because they do not know how to use. Some teachers have difficulties using them. These kinds of teachers should have training or improve themselves not to have problems against students. It would be absurd not to use technology in math class while these opportunities are providing to us. I am aware of them and I am sure to use where it is needed and how it should be done in my math classes.

     The reason of why this question is asked us is increasing our awareness of technology integration in mathematics and this was beneficial for me to think about my thoughts in this context. Before I write this assignment, I thought about which technological applications and tools we used in past and now. Then I realized there are so many programs that I do not know and I planned to use some of them in my classes in the future.