Nov 4: Euclid's Elements and beauty

Euclid and Euclidean geometry were concepts I first encountered during my undergraduate studies at UBC, particularly in my Euclidean Geometry and Discrete Mathematics courses. It was not until later that I realized how foundational Euclid's work had been to much of the mathematics I had studied for years. It is present in topics like trigonometry, geometry, algebra, number theory and calculus. As a result, Euclid and Euclidean geometry remain highly relevant in the mathematics domain. 

The Elements are also appealing for reasons outside of math applications. First, the framework of Euclidean geometry aligns with our natural intuition for visualizing space in one, two, and three dimensions using points and planes. Second, Euclid’s postulates form an axiomatic system that follows a clear and logical progression that mirrors how many of us reason today. Finally, Euclid's Elements have influenced many fields beyond mathematics, including art, architecture, astronomy, and music, highlighting their impact on daily life.

Many people may find the Euclidean postulates beautiful because they enable the derivation of profound theorems from simple, intuitive facts. The use of geometric shapes and symmetries in these theorems also adds to their visual appeal. Lastly, the logical flow of the proofs makes them easy to understand, further enhancing their charm.

Oct 16: Dish puzzle

To solve the dish problem, I reduced it to a smaller problem. To avoid using algebra, I utilized a number line to understand the pattern behind the distribution and total number of dishes. In the end, I concluded that the distribution of dishes would be the sum of all (number of guests)/(handout pattern). This would equal the total number of dishes being handed out. Since dishes are integer values, we also must consider the floor of the ratio. Applying this strategy to the larger problem, I got n/2+n/3+n/4=65 with n being the number of guests. After computing, we get n=60. Therefore, there are 60 guests. 

I believe introducing examples and puzzles from ancient history can be exciting and engaging for high school students because these concepts may be unfamiliar. Often, students disengage from problems when they are too difficult for their current skill level, rather than due to the context or story of the problem. Thus, the novelty of ancient problems may actually enhance their engagement. I also think students may find many ancient puzzles simpler than modern ones. This is because they involve tangible scenarios and originate using non-modern methods. Consequently, students may start puzzles with confidence due to their reliance on modern strategies and learn valuable lessons about their preconceived notions when they encounter challenges.

The narrative or context of a word problem is crucial for engagement. If the story does not capture the audience’s interest, they are less likely to invest time in it. Furthermore, during an age where entertainment is constantly accessible, it is important to ensure that problems are relatable, fresh, creative, and clear to maintain engagement. Lastly, from the FPPL, we know that learning is embedded in histories and stories. Thus, having story and imagery in word problems or puzzle is crutial.

Assignment 1 Reflection

My group initially chose this ancient question because of the familiar feelings that arise when a mathematician encounters "Pythagoras." I believe the extensive education around his theorem and the application of right triangles in nearly every high school classroom provides a sense of comfort. However, this familiarity created a bias that made it challenging to approach the ancient Babylonian methods with fresh perspectives. It took me some time to grasp their normalization techniques (such as normalizing an adjacent side versus the radius) and their concept of reciprocals (where their reciprocals multiply to 60 instead of 1). I think that if we had been introduced to the history of mathematics and various ancient methods at a younger age, it would have been easier to appreciate multiple approaches to problem-solving.

This project also highlighted the Eurocentric biases that persist in our historical narratives. I knew that many contributions from ancient Chinese and Indian mathematicians have gone unrecognized, but it was eye-opening to realize this issue extends beyond those civilizations. As a result, it is disappointing to see the lack of progress made in renaming mathematical discoveries to credit their rightful origins in textbooks and materials. As a teacher, I can play a role in this renaming process by educating my students about the rightful attribution of these discoveries. This approach may contribute to a more inclusive understanding of mathematics.

Lastly, I reflect on the evolving nature of mathematics. Before exploring extensions, I did not realize there were very modern solutions, like the use of complex numbers and planes, for addressing Pythagorean problems. This experience reminds me that mathematical methods are not fixed; they are constantly evolving. It was also satisfying to see how our problem, and many others, was solved using graphs, models, and diagrams instead of algebra. I find this shift away from algebra may help students build and trust their intuition more often.

Slides: https://docs.google.com/presentation/d/1Zh3KI2XFcd55FQTPpcuDYn1CV4C7EKXrHwDTp-TZSes/edit?usp=sharing

Oct 2: Market scale puzzle

The four values of the four weights are 1, 3, 9, and 27. My strategy for solving this question included a combination of reducing the problem into smaller parts and guessing/checking. To satisfy herbs of weight 1g-4g, I reduced that 1g and 3g would suffice. Then, in order to satisfy 40 grams, I concluded that the remaining weights should add up to 36 grams. I began testing, starting at 5g and 31g for my third and fourth weight. As I encountered gaps, I would systematically increase the third weight and restart my testing process. Ultimately, I landed at 1, 3, 9 and 27 as the answer. I was shocked to notice that these were the powers of 3. 

Using the theme of powers, I immediately tested the powers of two for the second question. It also related to our previous decisions in the garden about Egyptian multiplication. We learned that any number may be represented using powers of two. Furthermore, 1+2+4+8+16=31 which met the criteria for the one scale problem. Thus, the five weights for the second problem are 1g, 2g, 4g, 8g, and 16g. 

I may extend the problem by getting students to explore how adding weights would increase the quantities of herbs we may weigh. Does this number go up at a constant rate? Exponential? Or something different? One may also have fractional weights consistent with the base system (i.e. ⅓, 1/9…) to examine how precise one may get with these quantities. Could we construct any fraction using base 3 fractions? Why or why not? Lastly, students may explore how the base changes when one adds or subtracts states of the question. 

Overall, this puzzle connects with my prior knowledge of binary numbers. Specifically, it relates to the fact that any number may be represented using a binary system. Since a third state is added to this puzzle (left, right, or not included), we may similarly use a base 3 system. 

Sept 25: Reflection on word problems

Word problems have become an element of math assessment that most students dislike. Why is that? First, the topics of word problems may be too dull. In hopes of making math more applicable and related to the students, word problems have heavily become based on real-life scenarios that, for the most part, have nothing to do with the interests or concerns of a teenager. Often, these problems, like calculating mortgages or mixing solutions, feel forced and uninspiring. Hence, students are unmotivated to tackle them. Therefore, I would incorporate more elements of storytelling and subjects that relate to teenage life so that word problems become an element of play in the lessons. That way, students can exercise their creativity in math. I would also stray away from using word problems in purely assessment settings. For example, word problems may be used as brain breaks or during free time as collaborative activities. This may relieve the pressures students feel when encountering word problems, thus generating more positive sentiments around them. 

Overall, my opinion of using word problems as a form of creativity and play stems from the ancient genre from 4500 years ago. As discussed, word problems may have been a method to test and push the understanding of ancient civilizations rather than strictly practical applications. They also may have been used in social settings as forms of debate, connection or discussion. From this idea, classes today should also incorporate word problems to push student imagination and engagement rather than purely real-life applications. 

Sept 25: Egyptian Division

 

Sept 23: Ancient Egyptian Surveying

The article on Egyptian Surveying unpacks some of the methods and units of measurement utilized by the Ancient Egyptians. It confirmed some of the observations we discussed in class regarding the mural. These included using the rope and knots as a measuring tool, and measuring fields for tax and reporting loss purposes. While similar methods are still used today, I was astonished at their units of measurement. Egyptians are known for their precise calculations and infrastructure. However, according to the article, “the basis of the ancient Egyptian math unit length was… the length from the elbow to the tip of the middle finger” (2). Since this measurement may vary from person to person, I suspect there may have been occasional variances. For example, if I were to measure an object using my arm as a cubit, this may be inconsistent to another individual's arm. Despite this potential irregularity, the Egyptians were able to construct intricate, precise structures such as their tombs and pyramids. 

Two questions arose as I read the article. The first was regarding their techniques on right angles. The article proclaims that there is no evidence of the Egyptians using the special cases of the Pythagoras theorem during their measurements. Instead, they may have used as set squares or rhomboids to construct right angles. As a result, I wonder how this approach would be applied in larger-scale projects. Did they have to build large squares for each right angle calculation? The second question was concerning the significance of astro-surveying in modern times. In ancient times, the use of the sun and stars was very influential in calculating components like orientation, angles for large structures, and time. This may be because of the lack of light pollution, which allowed celestial objects to act as useful tools. Consequently, I wonder how prominent these techniques are in modern constructions. Given the increase in light pollution and development of other technologies, how often is astro-surveying done today?