The Dishes Puzzle

• Initially, I approached the problem by considering a small group of guests. For a group of 4, I found that there could be 6 dishes of rice (using "4 choose 2"), 4 dishes of broth (using "4 choose 3"), and 1 dish of meat (using "4 choose 4"). Adding them up gives 11 dishes, meaning each group of 4 would have 11 dishes. Since there are 65 dishes in total, I estimated there could be about 6 groups of 4, which suggests there are 24 people. However, this approach falls short because it doesn't account for guests who eat "across groups," meaning the guests can't be neatly divided into groups of 4. This led me to shift toward an algebraic approach.
• I then assumed a fixed number of guests and divided that number by 2, 3, and 4 to represent the number of rice, broth, and meat dishes, respectively. Adding them together should equal 65. Letting "a" represent the number of guests, the equation becomes a/2 + a/3 + a/4 = 65, which gives a = 60.
• Using examples, puzzles, and mathematical histories from diverse cultures can make a significant difference in engagement. This problem was especially engaging for me because I could visualize the setting—a bustling local restaurant with only a few items on the menu. I thought about how the cook might have gathered the information. With this context, I found myself more invested in the math. I also appreciated seeing a math word problem from a non-Western culture, as these are valuable resources that offer unique insights into different ways of thinking mathematically.
• The story and imagery in a word problem or puzzle are crucial. When the imagination is sparked, I explore different ways the problem can be interpreted and solved. The only insight I have is that, if I give students a puzzle with a specific answer in mind, I would let them know upfront to limit their imaginative scope and help them focus on the puzzle’s main point.

Reflection - Solving ancient puzzles in ancient and modern ways

TsáKtalay’pa and I tackled the proof of the formula for volume of a pyramid: V=1/3*base area*height, in ancient and modern ways. The original problem suggests investigating through two special cases, dividing a cube into six congruent symmetrical pyramids, and dividing it into three congruent oblique pyramids. We argued that the Ancient Egyptians likely derived the formula not from geometric dissections of a cube but through practical building methods. They likely began with a frustum for structural stability and efficiency of materials, and continued building upward to form a complete pyramid, perhaps aiming to connect with the stars. For the modern proof, the intuitions from these cases were helpful. We provided clay models for our classmates to explore the two divisions of the cube, along with paper templates to visualize the second case of three asymmetrical pyramids. We extended the discussion by introducing the Mayan pyramid, El Castillo, which resembles stacked square prisms and provided intuition for the formal proof using integration. This comparative approach enriched our understanding of the formula, blending practical, historical, and mathematical perspectives.

I learned a lot from both the process and content of this presentation. It sparked my curiosity about formulas I had previously taken for granted. Taking the time to explore and question them helped strengthen my relational understanding and appreciation of math, making me feel more prepared as a future math teacher. Understanding these concepts through ancient methods added historical context and narrative to the formulas, enriching the learning experience. I especially enjoyed the "extension" part of the presentation—not only was the research helpful, but it also encouraged me to connect and expand on different areas of my knowledge.

It was also really inspiring to see my classmates' presentations. I learned a great deal from this project, and I plan to adapt it into a class activity in the future.

Market scale puzzle + Connections

 

The four weights have to be 1, 3, 9, and 27. I tried working it out from 1 to 9 using different numbers first, then I realized the first 3 weights could be 3 to the powers of 0, 1, and 2. Then I tried 3^3=27 and it worked out perfectly. I don't know why these 4 numbers worked, and there seemed to be a unique solution to each value, but it was so satisfying.

If it were a one-pan scale with 5 weights, base-2 numbers with exponents 0, 1, 2, 3, and 4 seem to work! 1 + 2 + 4 + 8 + 16 = 31.

Riddles like this are so fascinating, because it shows patterns in Number Theory, and the fact that there might have been actual uses of it in the ancient world. For students to understand this idea and problem, I think the process that I went through - trying out with small numbers, recognizing pattern, expanding on the pattern, then deductively explaining how it works using what they know (though I'm still stuck on the last step) - is a good approach to new problems. The uncertainty and trial-and-error process is an entry point.

I don't know much about number theory, but there is something about the base 3 coinciding with the 3 possibilities in this problem. In other words, every weight is either not used, or used once for either the same pan or the other pan (3 possibilities). If I represent any herb weight value as N, 

N = 1a + 3b + 9c + 27d, with a, b, c, d belonging to the set {-1, 0, 1}

The above representation reminds me of linear algebra. But again, I still haven't figured out why!!

How would(n't) I use word problem in my own teaching?

The class discussion on Monday challenged my initial thoughts about the need for applicability in word problems. Initially, I believed that prioritizing real-world relevance meant using problems drawn from current events, up-to-date statistics, and relatable daily challenges. However, the discussion shifted my perspective. Word problems can instead focus on using mathematical tools for modeling, recognizing the patterns and beauty within math, contextualizing formulas, and humanizing math as a discipline. After all, problems in real life often appear as narratives, not simply as equations to "solve for x." This realization has made it clear to me that abandoning word problems altogether is not an option. The real challenge is in how I approach them.

Take, for example, teaching the Pythagorean theorem. A word problem could involve a carpenter needing to cut a right-triangular tile to fit an exact missing piece—or it could be a playful, imaginative story with the same core concept. It could ask students to validate the theorem by testing it with other triangles or exploring whether it holds true with different shapes. Students could also put themselves in Pythagoras' shoes, exploring the patterns as he did to understand his reasoning and how he developed the theorem. Additionally, they could be challenged to apply the theorem creatively, perhaps by designing a birdhouse, art project, or game. They could even create their own word problems and challenge their peers. In other words, my definition of word problems has expanded to encompass a variety of approaches, from incorporating humor and storytelling to fostering creativity and hands-on application. This broader view would hopefully enhance student engagement and their overall attitude toward math and word problems.

Response to Barnard (2014)

    Two questions that arise from Barnard's (2014) insightful article on "Surveying in Ancient Egypt" are: (1) Since "the basis for the Ancient Egyptian unit of length was the cubit, the length from the elbow to the tip of the middle finger," whose forearm was used to establish the cubit? and (2) Was Eratosthenes's calculation of the Earth's circumference, which implied the Earth was spherical, considered a radical idea at the time? The first question intrigues me because the answer could reveal which person, class, or gender held influence in defining standards—similar to how the "foot" was once based on the length of an English king's foot rather than the population average. Given that the tombstone mural depict people from different social classes, including children, participating in surveying, I wonder who had the authority to have their arm served as the standard unit of measurement.

    The second question is similar in nature to the controversy sparked by Galileo Galilei's 17th-century discovery that the Earth orbits the Sun. Did Eratosthenes’s conclusion that the Earth was spherical, with a measurable circumference, challenge any dominant religious or cultural beliefs, such as a concept of heaven existing above a flat Earth?

    What surprised me most was the Ancient Egyptians' understanding of the irrational number √2, which they estimated as nearly equivalent to 19.5 fingers (again, whose fingers?). Despite lacking precise tools, they approximated a value remarkably close to what we now know as √2. This raises an interesting question: have humans always been drawn to geometric forms like the square, leading to a natural curiosity about the length of its diagonal?

Response to Gerofsky (2004)

    As I mentioned in class, Francis Su's book "Mathematics for Human Flourishing" has reshaped my view of word problems, which often feel disconnected from real-world applications. Su’s idea of calling them "world problems" resonates with me, emphasizing the need to make them more relevant and practical. However, I was surprised by Susan Gerofsky’s "A Man Left Albuquerque Heading East", where she suggests that this disconnect might trace back to Babylonian mathematics! It made me wonder: has math education always been this way, or is it only in modern times that we’ve placed such a strong emphasis on real-life applications? Are we critiquing word problems through a contemporary, pragmatic lens, even though they’ve been integral to math instruction for thousands of years?

    Gerofsky (2004) references the Danish Assyriologist Jens Høyrup, who suggests that the Babylonians “had no other discourse available other than that of practical problems to express abstract mathematical ideas.” This is fascinating because I had never thought of word problems as a medium for expressing abstract math concepts. It’s similar to describing a real-world scenario today, then using math to model and solve it. In a way, word problems solve the challenge of not having precise measures or tools, serving as a bridge to understanding abstract ideas.

    I also found these questions thought-provoking: "Are word problems used primarily to train students in the use of methods without necessarily providing an understanding of those methods? Are problems chosen simply to illustrate the "methods at hand"?" (Gerofsky, 2004) The focus on “method” here is key. If the goal of word problems is to understand the use of methods, then I’ve definitely gained a lot from them.

45 Times Table in Base 60