GLM2 Task 4

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Western Governors University *

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OOT2

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Mathematics

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Apr 3, 2024

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docx

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Uploaded by mrapoza

GLM2 Task 4 A1. The standard that I will be using for the activity is [PC. G-GPE.A.3.a] - (+) Use equations and graphs of conic sections to model real-world problems.*. The topic is expressing geometric properties with equations. The focus specifically will be on translating between the geometric description and the equation for a conic section. 2. Students will research and explore significant mathematicians that had an impact on geometry, in particular the Greek mathematician Euclid. Euclid in his work “Elements created the foundation of conic sections as a mathematical entity. In his work, he provided explanations and properties for different shapes including ellipses, parabolas, hyperbolas, and circles. These findings lead to the development of conic geometry and many of his discoveries and formulas are still used to this day. Another mathematician that students would have the option to research is Rene Descartes. Descartes continued with Euclid’s findings and pushed the field of conic sections to another level of understanding. Applying much of what Euclid found and applying it to various fields of study. His greatest contribution though with this topic is in the area of analytical geometry. Analytical geometry lead to geometric shapes being created through algebraic equations. His creation of the Cartesian coordinate system, revolutionized geometry and advanced the study of conic sections along with other geometric shapes and objects.

2a. The activity that we would do with this topic would be a multi-day activity. The first day would be the introduction of our topic of conic sections. This would include introducing all four types, hyperbolas, parabolas, ellipses, and circles. Once we have introduced the topic, we would then dive into the history of conic sections starting with Euclid in Ancient Greece, and his contributions to the topic. Once we have explained the contributions of Euclid, I would mention how other mathematicians have had a heavy influence in the math we study today (examples would be Descartes, Apollonius, Euclid, etc.). Students would then be separated into groups and be tasked with researching an assigned mathematicians’ life, how they contributed to the topic we are currently learning (conic sections) and the impact that their work has on the world today. I would provide students with some preliminary research on their mathematician providing handouts summarizing their life and contributions. Once they have studied the impacts their mathematician has had on the world I would give each group a couple equations for ellipses and task the group to find the major and minor axis length along with the center of the ellipse. While students are working on the equation I would have students use both historical and modern methods to find the information they need from the equations I provided them. While they are working I would suggest to them to discuss and try to use the strategies that the mathematician that they just learned about would have used. While the class is working the teacher would walk around and monitor how the students are doing, facilitating conversations, and providing help with their work wherever needed. Once everyone has finished, the teacher will go through the assigned equations with each group, first having each group explain how they got their answers, and then clarifying any mistakes or confusion anyone is having. Once

the equations have been reviewed, each group will give a brief presentation, overviewing their mathematician and sharing their research with the class. Primarily focusing on their life in mathematics, how they relate to the current topic of conic sections, and their contributions to the world today. Finally, when those presentations are done, the teacher will end the lesson with a class discussion on how these historical figures and many more impacted how we learn math today, and has the class think about what it would be like without these contributions along with getting the class to understand why these contributions are so important to our world today. 3a. Students are able to engage with the historical component of this through their research, presentations and discussions of the mathematicians. Through doing this they are exploring the history of conic sections and the contributions that are still seen in the math they are learning today. One example of this is looking at the mathematician Apollonius, and his contributions to conic sections. Through their research students will find that he essentially laid the groundwork to many of the properties of circles, ellipses, along with other conic sections. Students could see how his work is seen today in architecture that started with his findings over two thousand years ago. The strategy to have students try to solve these equations the way some of these mathematicians did along with solving it in the modern way, intertwines the historical content with the present and it helps build a deeper understanding for the students. Euclid is a great example for this as through his work you can see the way math has evolved from his time. This

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helps create a deeper understanding and appreciation for the content. Along with this understanding the real-world impact of conic sections through learning about their contributions is a significant way to show students why these topics in geometry matter and their real-world impact creates a sense of interest and relevancy for the students. B. It is important to include historical context into mathematics because it will help students engage in the lesson, which will then go on to help them retain concepts after the lessons have been completed. When students learn about the historical figures that helped to mold modern math along with the real world impact these mathematicians had and the real-world applications that these topics have, students will have more of a drive to learn the material and absorb the material. This allows lessons to have more problem solving and critical thinking elements which forces students to apply what they learned and not just memorize the information. In the article “The Role of History in a Math Class” the author talks about how historical context can help serve as a powerful tool to promote mathematical understanding and motivation among students. The article speaks about how often times students view math as a standalone subject that does not have much influence in the world around us. Through adding historical context, it gives real world meaning to the content students are learning. Along with this, it also shows the evolution of math, and the changes it has undergone over the centuries, this helps students understand how we got to the math that we learn today. It also makes it seem more approachable and conceptually, it helps them feel comfortable being challenged knowing that

the greatest mathematicians throughout time have been challenged at different points of their lives. The article also speaks about how incorporating history also creates a classroom where students are active participants in their learning, involved in class discussions, consistently asking questions about the material. This also helps them connect the content they are learning to real world examples. These attributes will only help to improve students understanding of content increasing their critical thinking skills along with building their desire and drive to learn the material. Adding historical context to your lessons is an incredible way to boost engagement and retainment in your classes. C1. The German tank problem was a situation during World War II, where the Allied troops tried to estimate the amount of tanks Germany was producing during the war. The Allies had to use probability and statistics to try to estimate this number, and they were attempting to do so on the limited information that they had. C2. The German tank Problem is a major turning point in statistics and probability. Dur to the limited information, statisticians used estimates using real world data. This was not a practice that had been done previously, thus creating inferential statistics. Statisticians had to make educated guess through the limited information that they had estimating the possibilities of multiple potential sets of data. This leads to innovations in inferential statistics such as the

maximum likelihood estimation which is still a popular technique in inferential statistics to this day. C3. We would start our activity on the German Tank problem by first introducing the topic. First with a quick video summarizing the problem and then with a quick handout, this way students with all types of learning abilities can grasp the concept. Once we review the historical event the activity will be presented with the assignment of attempting to solve the German Tank Problem through a simulation. Students will be given data sample of how many German tanks are currently in use, through providing serial numbers of each tank. Students will be tasked to estimate the total numbers of tank based upon the initial sample size, similar to what the allies did during the war. As a class, the teacher will demonstrate for the students how to calculate an estimate of the total numbers of tanks through using the maximum likelihood estimate. While showing this method the teacher can also engage in a discussion about the limitations of the method, along with obviously what calculations need to happen to create an estimate. Students after the demonstration will work in small groups taking the steps shown to the class and apply it to their sets of data. Once students have completed their calculations they will quickly present their answers to the class and discuss and show their calculations that got them to that answer. At the end of the lesson after everyone has presented the teacher checks in with the class and clarifies any confusion in terms of using the maximum likelihood estimate. The class is finished

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with a reflective discussion reflecting on how this strategy is used in the real world today, and for what scenarios this math could be used. 3. Through including the historical context of the German Tank Problem, students are able to receive some cross-curriculum learning, and develop a better understanding of a historic event. However, more importantly students see that throughout history statistics along with probability have been used in real world scenarios throughout history. This provides students with an understanding and appreciation that the material that we learn has real world applications. Along with this, with this problem specifically students are required to make estimates on the incomplete sets of data they have, there are other situations in students lives where this skill may be important. Along with this, this is another example of how history has helped to form certain fields of mathematics, and helps students understand the importance of engaging with the material.

References Katz, V. J. (2012). A history of mathematics (3rd ed.). Pearson Learning Solutions. ISBN: 9781256819493 Marshall, G. L., & Rich, B. S. (2000). The Role of History in a Mathematics Class. The Mathematics Teacher , 93 (8), 704–706. http://www.jstor.org/stable/27971554 Mathematics Standards | Common Core State Standards Initiative. (n.d.). Www.thecorestandards.org; Common Core State Standards Initiative. Van de Walle, J. A., Karp K. S., & Bay-Williams, J. M. (2021). Elementary and middle school mathematics: Teaching developmentally (10th ed.). Allyn & Bacon. ISBN: 9780134802084

GLM2 Task 4

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