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
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
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